So, I have also played both of these games, and am now at the point where I can beat 2048 with reasonable probability.
This slide/move difference actually has massive effects on gameplay as it means there are situations you can get into while playing 2048 that are difficult or even impossible to recover from that you can easily extricate yourself from while playing Threes. To make up for this, the way you have to match 1s and 2s together (as opposed to simply matching the values, as you can in 2048) makes organizing the lowest level of the board slightly more complicated.
I frankly bet if we asked the developer of Threes about the 1st and 2st thing he'd say he toyed with the idea of a game where the matching was more direct (as its pretty obvious consider it) and realized that it felt "too easy": the only thing making 2048 continue to be difficult is the lack of explicit control over the tiles; though, that also makes it easier, as you can move items much more quickly around the board without more clutter appearing.
I would say this is similar to how in Tetris Attack you can horizontally swap any two tiles, but in Bejeweled you can swap tiles vertically or horizontally, but only temporarily: they have effectively the same mechanic (you need to match tiles of similar color), the same overall physics (items drop to fill in gaps), the same kinds of tactics, but the amount of control you have over the game board and what moves you have are quite different in feel due to the seemingly minor control changes.
(FWIW, I find Tetris Attack and 2048 "fun", and I find Bejeweled and Threes "infuriating". This is likely somewhat to do with the fact that I find Tetris Attack and 2048 "easier" than Bejeweled and Threes, but I would hope that it is more to do with some of the things I really enjoyed about Tetris Attack--the speed of movement, the building of structure, and the intricacy of "skill chains"--not being tactically relevant in Bejeweled, and in the case of Threes that the 1s and 2s are randomized in such a way where I often feel "this game isn't even winnable: I have a board full of 1s... this isn't even fair", which is a situation fundamentally impossible in 2048.)
