While to my knowledge, it isn't perfectly spelled out in-game, a reasonable hypothesis follows (basically this whole Answer is major spoilers):
The Chosen of the Dead Three come up with a scheme to steal an ancient Netherese artifact (The Crown of Krasus) and use it to dominate an Elder Brain (aka, the Netherbrain) to do their bidding. They do so. The Nether Brain does not like being magically enslaved, and begins to plot an escape.
The Emperor was a mindflayer operating outside the purview of an Elder Brain (and, in fact, had spent literal years waiting for an opportunity to escape). When its organization ran up against Gortash's influence in Baldur's gate, its true nature is discerned and, thus revealed, Gortash and co. bring The Emperor under the Netherbrain's thrall.
The Netherbrain, after dominating The Emperor, realizes two things: First, The Emperor will go to great lengths to avoid being dominated and enthralled. Second, this is an opportunity, as it has already realized that if one of the netherstone wielders is slain, that will weaken its slavers' control and eventually allow it to escape the domination.
This allows The Netherbrain to conclude that The Emperor is likely to facilitate such a killing, in order to secure its own freedom (or, potentially, claim the Crown of Krasus for itself). One problem: The Chosen dominate the Netherbrain, and the Netherbrain dominates The Emperor. If the Chosen forbid it from acting against them, The Netherbrain's thralls are transitively bound to the same rules.
So The Netherbrain lets slip that the Astral Prism contains a power that allows one to resist the domination of an Elder Brain, and subtly insinuates that The Emperor would be the agent best suited towards retrieving it. The Chosen fall for the gambit, and The Emperor successfully acquires the Astral Prism.
This temporarily returns The Emperor to independence, but The Emperor is desperate for a more permanent solution. But The Emperor is but one monster -- they're still surrounded by Absolutist forces, and, it needs to hold onto the Astral Prism (made more difficult by the Orpheus' honorguard). In short, it needs help.
Remember - Illithids are intrinsically manipulative. [Cite: in game texts]. The best way it knows to "acquire help" is actually spelled "coerce".
So The Emperor makes its way to the recently captured and tadpoles as many as it can (remember, at this point all the other Mindflayers on the Nautiloid are still under The Netherbrain's control -- hence the illithid corpses in the opening cinematic).
Now that they have a "sword of Damocles" above their heads, all that's left to do is escape from the Nautiloid and begin puppeteering its new "allies". Since Gortash directed The Emperor to pilot the Nautiloid (as per his journals), this is actually quite easy -- allow the gith to catch up and down the ship, use its psionics to protect its newly tadpoled "friends" during the crash, and retreat into the Prism where it can suckle Orpheus' power to maintain the status quo.
So why Tav specifically? Convenience, really. Wrong place / wrong time. The Emperor needed someone who could fight against The Chosen and be manipulated into pursuing those goals. Tadpoling the cast allowed it to achieve both those ends.
One thing that doesn't fit nicely into the above understanding is how the Sharran artifact retrieval squad fits into things. Shadowheart claims that they steal the artifact from the gith, Gortash tasks the Emperor to do likewise, and both can't be true. Then again, Shadowheart's memories are all over the place, and a little memory manipulation seems perfectly within The Emperor's wheelhouse.
The other strange wrinkle is that if you are playing The Dark Urge, the opening cinematic scene isn't canon for you. Specifically, TDU is tadpoled before boarding the nautiloid.
Which leads to an open question: Perhaps the opening cinematic has been mildly retconned? The Emperor's eye color is wrong, for one, even if the cowl otherwise identifies it. Consider too, that the cinematic predates the game's final release by ~3 years, and that a full 3d cinematic like this isn't nearly as easy to change as writing or dialogue.
So in short: the cinematic isn't perfectly aligned with the implication, so it's also entirely possible that the premise of this question is false, and The Emperor is not the one who does Tav dirty. It's ambiguous, and there's yet to be anything to conclusively verify it, one way or the other.
Raven Dreamer’s answer gives a good overview, but if you’re interested in an exhaustive list of changes, one is available via the independent Baldur’s Gate 3 wiki project. The main article covering changes to basic rules like initiative, healing, resting and more is “D&D 5e rule changes”, and there are also further articles for changes to races, classes, spells and feats. These articles all assume some familiarity with standard 5E rules. These pages are also updated when patches make further changes to the rules used by BG3 (especially ones which fix bugs preventing a rule from operating as it does in the tabletop game).
I would add that there are some changes that are not obvious but have a big impact on the game. As one example, initiative is not a standard Dexterity check, but instead uses a d4 (though the Dexterity modifier is still added). This makes it more likely player characters will end up next to each other in the initiative order, and such characters effectively act simultaneously, able to take their actions and movement in any order they like. Like many of the changes to a basic rule, this has many knock-on effects, such as modifying how many bonuses to initiative work (advantage on initiative rolls is generally changed to a flat bonus of +3 to +5, for example).
A few other significant changes which surprised me and affect power level:
- Ranges for weapons and spells have been reduced across the board (usually to 60ft/18m), and combat generally takes place in smaller areas with a lot of opportunities for vertical movement. Being at least 10ft (3m) above an opponent can grant a bonus on ranged attacks, and likewise being below them imposes a penalty. (By the way, the game can use metric measurements, using rounded approximations of 5 feet = 1.5m, and 1lb = 0.5kg. As an Australian I love this.)
- Cover is not implemented, but objects and creatures can block line of sight.
- Prepared spellcasters can change their list of prepared spells instantly at any time outside of combat.
- Ritual spells don’t take longer to cast, they just don’t use a spell slot unless cast during combat. Some higher level spells are now rituals, including speak with dead.
- As you noted, most spells and effects with a duration longer than 1 minute now just last until the party’s next Long Rest.
- Most potions can be thrown (as an action) in order to make them splash on the ground and apply their effect to multiple characters. This includes healing potions.
- By default the game has a “Karmic Dice” option enabled which prevents streaks of very high or very low rolls. This always acts in favour of the character or creature rolling the dice, and results in shorter more dangerous combat, since everyone is more likely to hit and do higher damage. This gives spells with saving throws a disadvantage vs ones with attack rolls, since the target rolls the dice.
- Humans and half-elves have revised traits to make up for losing their additional Ability Score bonuses (since everyone gets a choice of +2/+1 to any Ability Scores). This includes a “Civil Militia” trait which grants them proficiency in various polearms and shields.
- Spell components (including foci) are generally ignored, so a human wizard can equip a staff and shield and still cast spells. Being silenced (for example by the silence spell) does prevent spellcasting, which along with the vocalisations when characters cast spells suggests an implicit verbal component, but a small number of spells are explicitly called out as being usable while silenced.
- The AC bonus from a shield equipped in the melee weapon set is applied even while using a character’s ranged weapon set.
There are many more, but those give you an idea about the extent of the changes.
Best Answer
Before you engage, you can separate your characters by either dragging their portraits on the left side out of the row (in which case separation is indicated by the chain breaking), or by right-clicking their portraits and selecting "Ungroup" There should now be more space between the portraits. You can also Toggle Group Mode, i.e. (un)separate all party members, by pressing G.
When they are all separated, they won't move towards each other on their own accord, and you can guide them separately.
This is also extremely useful for scouting and disarming traps.
Next thing to make sure of, is to not end the turn of the initiating character until all other party members are actively participating (unless you want other members to join in later, of course).
For me, it doesn't actually matter whether I engage the enemy in Turn-Based mode or in Real-Time mode (I wonder if it depends on the Difficulty level, as I'm playing on Balanced difficulty, but I wouldn't know why that would matter).
Once the enemy party is alerted, one or more of them might turn and notice party members that you had hidden behind them. They will still have the Surprised status, allowing for that initial non-challenged round, but if you want them to remain hidden, be aware of this.
The advantage your party has over enemies is also somewhat dependent on their Initiative, in the sense that once they enter combat they might end up later in the row, with the consequence that they could be attacked (by enemies with a higher Initiative who lost their Surprised status) before they are able to act in the second round.
The steps that always work for me (trying to be as thorough as possible):
Your other characters will remain undetected (unless, again, the changed lines of sight inadvertently caught them and they engage as well).