What would you like to do? = What would you like to do now? or What would you like to do <at a particular time> "We have today off from work. What would you like to do [today]?" What do you like doing = What do you like to do at any time. "What do you like doing?" "I usually like to play tennis. I also like to swim."

1-What would you like to do...is like saying what do u want to do... ( this afternoon, this weekend...[at a particular time]) 2-What do you like doing... ( in ur free time ). in this sentence u are asking someone generally what does he do....., like asking him about his habits !! the tense here plays an essential role, the present simple describe habits and routines. hope it helps :)