They are QUITE different
They serve both a different purpose, AND contain different content.
Newbie Set vs. Legal Document
The Basic Rules were released prior to the SRD, and are designed to provide a very limited set of gameplay material in order to give players a taste of D&D 5E before they purchased it.
The SRD is a legal document intended to provide all of the 5E rules and material that Wizards of the Coast consider to be Open Source material that is legal to use under the Open Game License (OGL). You may read the details of this license at the top of the actual SRD pdf
In other words, if you want to create and sell material for D&D 5E, then you may only use material that exists in the SRD or material that you invent.
For example...the iconic D&D monster The Beholder does not exist in the SRD. Therefore, if you want to create content under the OGL, you may not use a Beholder in that content.
Differences in Content
The SRD and the Basic Rules contain dramatically different amounts of Content.
The Basic Rules only includes content for 4 classes: Cleric, Fighter, Rogue and Wizard, each with only a single sub-class (Life Domain, Champion, Thief, and Evoker). Only 4 races are offered: Dwarves, Elves, Humans, and Halflings.
The SRD, on the other hand, contains every Race and Class from the PHB, with at least one sub-class per Class. Additionally, there are several Monsters included in the SRD that are not in the Basic Rules, such as the 'Tribal Warrior.'
Going the other direction, the SRD includes precisely one Background: The Acolyte. The Basic Rules contains 5 backgrounds.
Conclusion
I suppose the best way to show the difference is to simply share what Wizard of the Coast says about them...
Regarding the Basic Rules...
The Basic Rules is a free PDF. Anyone can download it from our website. We want to put D&D in as many hands as possible, and a free, digital file is the best way to do that.
And the SRD
The Systems Reference Document (SRD) contains guidelines for publishing content under the Open-Gaming License (OGL).
Maybe.
The AL Player's Guide and AL DMG don't address which rules to use outside of setting minimums and outlining the "+1 rule." So we turn to the AL FAQ (version 7.1 as of this writing) which contains this relevant guidance on p.2:
What Rules Do I Use?
As a D&D Adventurers League Dungeon Master, you are empowered to adjudicate the rules as presented by the official materials (PHB, DMG, MM, etc.).
So it has to be "the official rules," and PHB, DMG, MM are examples of such rules.
The next page makes a bit of an argument against, in that it lists twenty-four separate sources and doesn't list D&D Beyond, but arguments from omission are inherently weak, in my opinion.
That said, since the differences between D&D Beyond and the PHB, for instance, haven't been issued as errata you're "safe" going with the legacy wording.
Proceed with clarity.
In a few months we should be seeing a set of AL documents in the v.8 form which one hopes will make some mention of D&D Beyond. Until then just be clear with your players/GMs what you're using. Either "I don't see D&D Beyond the list so I'll stick with listed sources" or "it's an official WotC product which is what they say to use" certainly strike me as legitimate positions. As usual: what is important here is probably not so much arriving at a "correct" conclusion as is arriving at a shared conclusion.
Best Answer
(Preface: The below answer was initially written before WOTC's acquisition of D&D Beyond on 2022-04-13. I assume the acquisition would not make my conclusion of 'treat the digital sourcebooks as official' less valid.)
I believe at least the digital versions of the sourcebooks themselves (currently listed here) should be treated as an official rules source roughly on par with the books.
Firstly, WOTC considers D&D Beyond to be an official digital toolset for the game (though, until the acquisition, D&D Beyond wasn't made by WOTC directly, but licensed through them):
Secondly, the books on D&D Beyond are considered to be 'digital sourcebooks', which, on their own site, are defined thusly:
As they are considered a 'completely digital version of one of the published books', I'd say they're as official-source-worthy as said books.
In addition, BadEye (Adam Bradford, D&D Beyond product lead at Curse at the time of posting) also calls D&DB an official source, mentioning that their site is meant to be kept up to date with the rules as new changes come in:
Ultimately, due to the above, I feel D&D Beyond's version of the sourcebooks should be considered as official as the books themselves. If there's a discrepancy between their content and the most recent errata/printed version of a book, then the book (plus relevant errata) is the 'most canonical', sure, but I don't think the potential for that kind of mismatch/data entry/out of date error would make the rules from the site overall considered 'unofficial'.
Disclaimer: The above only applies to content on D&DB from the official sourcebooks. D&DB also hosts various things that shouldn't be considered official, such as usermade homebrew, forum posts, and unofficial sources such as the Blood Hunter class from Critical Role/Matthew Mercer- these are usually kept separate in the site's navigation. The SRD & Basic Rules are also hosted there in a combined section, but even the WOTC-hosted PDF/printed versions of the SRD is considered unofficial:
In addition, applicable Unearthed Arcana content was once hosted there (for the UAs released in the range of Jan 8, 2018 to the discontinuation of UA on D&DB around August 2021), which should be treated the same 'officially unofficial' way any other UA content is treated.
While the parts of D&D Beyond that are aggregates of data from the sourcebooks (such as the "Races" section, the "Monsters" section, etc.) contain reproductions of sourcebook data, they seem to be slightly more prone to data entry errors- I'd recommend using the digital sourcebooks' version in the case of finding a conflict between the two.
(On the topic of "what's considered an official sourcebook?", @nitsua60's longstanding question/answer "Where do I find the “official” rules for D&D 5e?" is worth a read as well.)