(Note: If anyone wants to add specifics on crunch to my answer, or copy-paste my answer to use as the basis for a better answer, please do so. I don't know enough to be helpful!)
I presume your approach is this: You want to try to combine the 3.0 core rulebooks with the 3.5 SRD, presumably because you want to use the 3.0 books for source material and advice but the SRD for stat blocks and an authoritative system of rules. You want to know what will you miss, meaning what could trip you up because you missed the difference between 3.0 and 3.5. I'm going to explain from a GM's perspective, that is, that you'll be planning a campaign, and that you've familiarized yourself with 3.5 already.
In short, as long as you remember to always treat the SRD as the only source for crunch, you won't miss much. A very small amount of mechanical data is missing from the 3.5 SRD (such as the stat blocks for the few monsters that Wizards kept out of the SRD, and the PHB's XP table). You can use the old versions safely, when necessary, provided you're mindful of systemic differences (such as the change to Damage Reduction). It'll be easy to know when to be careful, though, because you'll be looking at a book instead of a website.
Let me dig in to each core rulebook for specifics.
Player's Handbook. Honestly, just tell the players not to trust the 3.0 PHB and you'll be fine. Everything that isn't crunch is common cultural knowledge at this point, such as knowing that fantasy dwarves are good miners and like to use axes. The XP table is all that's missing, I believe, and it didn't change.
Dungeon Master's Guide. I actually own the 3.0 DMG but not the 3.5 DMG, even though I haven't GMed 3.0 in years. In fact, I reference it all the time for my Pathfinder game, for its general tools and advice on how to GM. So I think you'll do very well without the 3.5 DMG and your players will never notice a difference, provided you always use the SRD for actual game mechanics similar material such as the magic item chapter. But, say, the table of 100 plot ideas will never be out of date, you know?
Monster Manual. Effectively you'll be ignoring it wholesale except for, e.g., reading the colorful monster descriptions aloud to the players, and that's safe since they didn't redesign any monsters in terms of concept (that I know of). There is the matter of the non-SRD monsters, but again, you'll know to be careful when using them, so you'll be fine.
Seriously, and according to its own texts, Wizards of the Coast says
The Rules Compedium Can't Change the Core Rules
I know it's weird. I know it sounds like nonsense. I know that when the core rules were published things like swift actions didn't exist. I have shed blood on the Internet battlefield between the Rules Compendium declaring free actions are only available on one's turn versus the Player's Handbook slightly vague addressing of the topic, especially in conjunction with immediate actions. I know. I know.
Wizards of the Coast Created a Flawed Paradigm
Here's the skinny: All of the errata documents have this Errata Rule describing Primary Sources
When you find a disagreement between two [Dungeons and Dragons] rules sources, unless an official errata file says otherwise, the primary source is correct. One example of a primary/secondary source is text taking precedence over a table entry. An individual spell description takes precedence when the short description in the beginning of the spells chapter disagrees.
Another example of primary [versus] secondary sources involves book and topic precedence. The Player's Handbook, for example, gives all the rules for playing the game, for playing PC races, and for using base class descriptions. If you find something on one of those topics from the Dungeon Master's Guide or the Monster Manual that disagrees with the Player's Handbook, you should assume the Player's Handbook is the primary source. The Dungeon Master's Guide is the primary source for topics such as magic item descriptions, special material construction rules, and so on. The Monster Manual is the primary source for monster descriptions, templates, and supernatural, extraordinary, and spell-like abilities.
Emphasis mine. So the most recent publication of the core rules--the 2013 editions of the Player's Handbook, Dungeon Master's Guide, and Monster Manual--are absolutely primary, even if topics within those texts were subtly changed, outright contradicted, or the subject of FAQ or game designer commentary by other texts between 2000 and 2013 before those texts' publications, and even if uncorrected errors remain in any of the most recently published core books that were corrected by other books before the core rule' republication.
Flaws other books set right may have been backwardly-uncorrected--or whatever Orwellian phrase you might want to use--by the republished core rules.
The Rules Compendium Asserts Its Own Primacy
The Rules Compendium's Introduction says
When a preexisting core book or supplement differs with the rules herein, Rules Compendium is meant to take precedence. If you have a question on how to play [Dungeons and Dragons] at the table, this book is meant to answer that question.
So while the Rules Compendium is meant to take precedence, it, officially, can't.
Why Reject the Rules Compendium?
Some folks enjoy playing by the rules--the rules are there, someone was paid to write them, and some (probably) paid to acquire them--, and having the core rules changed by a source and only that source makes them uncomfortable, like somebody's trying to pull a fast one. Some of the Rules Compendium's so-called clarifications are outright changes to the rules, and that bugs folks.
Examples of changes implemented by the Rules Compendium include...
- Charging through Hindrances: The PH says, "You must have a clear path toward the opponent, and nothing can hinder your movement (such as difficult terrain or obstacles)" (154) then goes on to define a clear path as empty of such things as opponents and allies. The RC allows the use of some skills to avoid movement hindrances during a charge (27).
- The Action Needed to Activate Some Magic Items: The DMG says that activating of spell completion and spell trigger magic items "is a standard action" (213). The RC, says, "Activating a spell trigger [or spell completion] item takes the same amount of time as the casting time of the spell that the item stores" (85).
- The Survival DC for Avoiding Quicksand: The original Survival skill check DC for avoiding quicksand is 8 (DMG 88). The Rules Compendium on page 103 increases this DC to 15 without commentary.
- Touch Spells and Threatening an Area: The PH is unclear on whether a creature who lacks the feat Improved Unarmed Strike, a natural weapon, or both threatens an area with a touch spell's held charge (PH 141-2). The RC puts forth decisively that "a spellcaster delivering a touch attack spell... count[s] as armed. Being armed in this way counts for both offense and defense. So a creature armed in this way can make attacks of opportunity, and such a creature doesn’t provoke attacks of opportunity when attacking" (16).
- Using the Skill Sleight of Hand: The rules for the skill Sleight of Hand are on PH 82. The RC maybe expands but possibly replaces this description on page 117, yet the RC excludes the entries Action, Try Again, Special, Synergy, and Untrained present in the PH. Some RC changes go unmentioned in the text yet appear on the nearby table (e.g. a creature suffers a -20 penalty when taking a move action to make a Sleight of Hand skill check). Whether the RC's Sleight of Hand skill description is to supersede or supplement the PH description is unmentioned by the text.
- What It Means to Be Hidden: Using the PH alone, the only effect of being hidden while in combat (beyond being undetected) is the possibility of combat ending to take advantage of a future surprise round. This is substantially changed by the RC's description of the skill Hide (92).
- When Daily-use Items Recharge: The core rules don't provide a general rule for a magic item (or, for that matter, any special ability) with 1 or more abilities usable per day to regain its uses of those per-day abilities, necessitating house rules. The RC addresses the recharge rate for daily-use magic items that don't otherwise provide them on page 86.
"This is Bizarre! Can I read more?"
Sure. Giant in the Playground forums' Curmudgeon is probably the most articulate and vociferous proponent of the Rules Compendium's flawed existence. His comments here summarize his stance well, and it's a good read.
Note: I'd like to make this answer a repository of links to questions and answers that demonstrate where the core rules and RC differ. You can add such into the Examples of changes... yourself or leave a Comment.
Best Answer
No other source book currently printed overrides the core rules (unless you want it to)
Other than the PHB, DMG, and MM, the only other source book that resembles a general rule book (as opposed to a book describing a setting, adventure module, or other additional content) is Xanathar's Guide to Everything (XGtE). And XGtE explicitly says in its introduction that it does not override the core rules:
In short, all of the rules in XGtE are optional rules. Taking an example from your question, the falling rules given in the PHB are quite simple:
That's it. The rules don't even say when you fall, they just say what happens when you do fall, and mostly leave it up to the DM to determine when falling happens. XGtE gives some additional (optional) mechanics to use for falling, none of which contradict or override the PHB's rule quoted above. In fact XGtE begins its section on falling by restating the PHB's rule itself:
Of course, even in cases where XGtE contradicts the core rules (one well-known example being magic item crafting), there is no question of precedence, because XGtE explicitly cedes precedence to the core rules in all cases by declaring that all of its rules are optional. For example, it's entirely up to the DM whether to use the DMG's crafting rules, XGtE's crafting rules, or something else entirely.
To summarize, there is no general rule for which books take precedence over which other books. Instead of defining a general rule that is external to the rule books, the books themselves tell you which ones take precedence (with the only current example being XGtE).
Non-book rule sources
Of course, there are other rule sources besides printed books, as well as sources of things that are not rules. Here is my understanding of how they fit in. (Thanks to the many commenters for bringing most of these up.)
Errata
Errata issued by WotC are corrections to unintentional errors, ommissions, and the like in a specific source, and as such always override the source they are correcting. They don't change the order of precedence between sources, so for example an erratum for XGtE would not override a rule in the PHB.
Basic Rules
Wizards of the Coast provides a free "Basic Rules" PDF that contains selected content from the core rulebooks (mainly the PHB and DMG, I believe). Unlike XGtE, which is explicitly positioned as a a companion to the core rules, the Basic Rules document is designed to stand alone, such that you could (in theory) play D&D 5e using only this document, albeit with a relatively limited set of options. As such, the Basic Rules do not declare any explicit relationship to the core rule books. Generally there should be no problems arising from this ambiguous precedence, since all the rules in the document are excerpts from the core rule books, so in theory there should be no conflicts, except where errata issued since the publication of the Basic Rules (dated November 2018) have caused the latest printings of the core rules to diverge. Obviously, in such cases, the errata and/or corrected printings of the core rules would prevail.
D&D Beyond
All printed sources are also available in online form on D&D Beyond. While D&D Beyond is run by a 3rd party, they have licensed the official materials from WotC. The content on D&D Beyond should always correspond to the latest printing of the corresponding book, with all official errata incorporated into the text. In other words, there are no new official rules presented on D&D Beyond; it's just an online source for the same rules that are already published in printed form.
It's worth noting that in addition to simply reproducing the content of each source book, D&D Beyond also collects content from multiple source books into single pages for specific topics. For example, the D&D Beyond page on the wizard class includes all the relevant content from the PHB as well as all the additional arcane traditions published in other sources (at time of writing, this includes the War Magic tradition from XGtE and the Bladesinging tradition from the Sword Coast Adventurer's Guide). Every piece of information on the page is reproduced from a source book, but there is no single source book that contains all of the information on this page. For the most part, this shouldn't matter, but if you do run into any issues relating to rules precedence on these multi-source pages, you should resolve the issue using the original sources.
Sage Advice Compendium
This is an official publication from WotC, but it contains official rulings, not additional rules. The goal is to clarify how the existing rules are intended to work, not change them. As with XGtE, this is explained within the Compendium itself:
Jeremy Crawford's tweets (and other public statements by WotC)
The Sage Advice Compendium further explains that it is the only source of official WotC rulings, and no ruling is considered official until it appears in the Compendium:
(Jeremy Crawford's tweets in reply to rules questions were previously considered official rulings, but that has changed recently. See here for information about the change.)
Unearthed Arcana
WotC occasionally publishes play-test documents referred to as Unearthed Arcana. As with other WotC-provided materials, these documents always clearly explain their relationship with the core rules. For example, the 2019 revision of the Artificer class says:
In general, official rules will never be published in Unearthed Arcana. Any UA that does become official does so by being published in an official source book, at which point the official publication should take precedence over the UA version.
Sage Advice D&D website
There is a website called "Sage Advice D&D" that is unaffiliated with WotC. It has nothing to do with the official Sage Advice Compendium. Among other things, this website collects rulings made in tweets by the D&D designers. However, as noted above, these tweets are neither rules nor official rulings.
(Note: I have no experience with Adventurers League, so some of the above may not be correct for AL.)