What is the point of a dresser?

It's like the clutter option in the game, in my opinion. There for aesthetics; decoration really.

You can put stuff on it. You can't do that with a closet.

I never have paid attention to dresser animations, but they opened in Sims 3, I know that. I remember sims falling over from pulling them open! :)

I agree with @asesina they are mostly there for decoration... but you can put clutter on them to make it seem more "realistic". I don't use them in my game but I do put them in my house builds.

They are a hold over from previous games, where they were used to access CAS. Now they are mostly decorative, there to fill in the empty space in bedrooms.

I hardly use them anymore since you can change clothes with mirrors. It opens CAS just the same. I only use walk-in closets in my builds but that's... for other purposes. :D

I only used them to put stuff on them to make the house worth more also too I guess, other than that there really is no point.

I still like and use them. They complete the look of a bedroom, and I still send Sims to them to 'plan outfit' or 'change outfit' because it's part of the immersion for me. I do sometimes change outfit on the Sim directly, but usually this is when they're out and about and I need a quick change. If I want to change make-up, I tend to do it on the vanity now. I use mirrors for freshening up, brushing teeth and practising speeches.

I don't see the drawer animation as a big deal, although it would be nice to have.

Right I still use them, same as the armories, make the husband change outfits out of the armorie, and the wife tends to use the bureau but they both share the closet.

I tend to leave them out of my builds in game unless the family has money to burn. I always have a mirror though because they are cheap, have all the same interactions and more on top.

Like @Dannydanbo said though, they often get used for other things, like filling alcoves below shelves, or as sideboards in dining rooms - great for party food.