We went to Tate Modern just after xmas - we'd planned to go to the British Library but it was closed. Tate Modern was an awful experience, it was packed and there's no way you can even begin to appreciate art when there are people shoving and milling around (and screaming kids running around waiting their turn on the slides). We all came out tired and grumpy and only chocolate could revive us.
A couple of days later we managed to get into the British Library - the London maps exhibition was fascinating even if we couldn't help but spend time looking for Hanwell (or Hanuell) on the maps that reached as far west as us. The permanent exhibition was really interesting as well (although I can only cope with so many illuminated manuscripts! The whole atmosphere was so much more pleasant than the nightmare of the Tate!