Oh my, where to begin. I have lived in both cities. People of done a great job of covering the "usual suspects" so I will add a few of my off the beaten path thoughts.
In London, I lived in Kensington. Holland Park (1 block from my flat) was one of my favorite escapes. (
http://en.wikipedia.org/wiki/Holland_Park). The Kyoto Garden was a favorite place to sit and read on a nice day. It is this amazing oasis in the middle of London. I was stunned one afternoon to find myself staring at a fox, not 10 feet away. During the summer they do Opera in the park.
In Paris, Laduree at 16 Rue Royale, 75008 Paris, France was a quick 3 block walk from my place in Paris. The millefeullie (what we call a Napoleon pastry) there is amazing. It is a bit touristy.... but still one of my favorites.
http://www.laduree.fr/v1/index_en.htm
I also enjoyed Rue Cler, which is a classic Parisian shopping street.
No matter what you do, it is hard to have a bad time in Paris or London
