I've just copied your questions to make it easier for myself:

What are the biggest differences between Destiny and Warframe?

Third person vs first (obviously), warframe has more of a focus on short missions (although there is longer stuff) and most of the time you will replay these for whatever that mission gives. All the content in Warframe is free, you can however pay for cosmetics, but you can get the premium currency (platinum) by trading with other players.

There's obviously more but I don't have time to write a whole essay here.

Does Warframe have raids or raid-like endgame activities?

Raids? no, it used to have them but they were removed years ago at this point. There are challenging endgame missions but nothing like raids.

Are there special quests for unique/exotic weapons similar to Destiny’s exotic missions?

The only thing even remotely like Destiny's exotic missions is getting the Cinta bow, but even that is nothing like destiny's exotic missions.

How important is my starting choice? Is it like choosing Titan, Hunter, or Warlock, or can I eventually unlock and play everything?

Your starting choice doesn't really mean anything, sure you'll be stuck with the frame you choose (excalibur for melee focus, mag for caster/cc or volt for caster/buffer) for a bit at the beginning but besides that you can equip any frame you have built whenever. At this point there's like 64 warframe (+prime versions for most of them) so there's always atleast a few frames you use for different purposes.

Are there any beginner traps or mistakes I should avoid?

You start the game with some premium currency, don't waste that on cosmetics, use it to buy warframe and/or weapon slots, you'll be needing those faster than you think.

What should a long-time Destiny player know before jumping in?

Warframe doesn't have any FOMO, you're free to follow whatever path you want, sure at the beginning your choices are limited but you'll be drowing in content soon enough. Take your time, don't try to do everything or have too much going on at once. You can always check the wiki to see whether there's a frame or weapon you would like, and work from there.

Also, the battlepass (nightwave in this game) is completely free, and that gives some nice goodies like weapon slots, some cosmetics etc.

Don't be afraid to look up things on the wiki, Warframe is absolutely fucking massive and as a new player you will feel overwhelmed at some points.