A British person should really be answering this because "keen on" is one of their expressions, not ours. (I'm American and the other person who answered is also from the U.S.) Here's what I *think* is correct, based on spending a lot of time talking with my British friends. I think "I'm fond of X" simply means "I like X." You can use it in almost any situation. We say this in America too. "Jane is keen on John" would mean that Jane likes John in a romantic way, just as fdmaxey said. You can also use "keen on ~ing" to say that you are eager or willing to do something. It's used to discuss specific plans or things you generally like doing. For example, "Are you keen on going to the cinema tonight?" Also, I think "keen on ~ing" is used more often in the negative, to talk about things you don't want to do. "I'm not keen on cycling to the office as it's going to rain." You could also use "not keen on" + noun. For example, my flatmate wants to have a party but I'm not keen on the idea.

You are usually keen on some idea or some activity. If you are keen on a person, it is almost always a person of the opposite sex, in a romantic way. 'Fond of' is not so intense. You could be fond of some food. You could be fond of a friend or relative, not necessarily in a romantic way.