Tate modern 
Tate modern is a big and has a collection that is large most of it is not on show.
i think if you are going to look around Tate modern it will take you a day.
Tate modern was an old power factory that was shut down and then remodelled as Tate modern.

V&A
the V&A is massive i think if you wanted to look at the the art work in there it would take you 3-4 days.
i like the V&A more as there is more verity then the Tate modern and that means there is something for everyone and your not going to get bored.

over all i find the V&A to be more help fun then the Tate modern as there is people all ways in the rooms and if you cant ready anything you can ask them or ask what they think of it. As for Tate modern you have to borrowed a electronic guide and that's £5 which i find is too much to say the gallery is free.