The fansite is the website itself, such as the Facebook or Twitter account, and the fanbase is the word for the people who follow that fansite! But these words aren't used very often, so don't worry about it too much!

A fansite is any website/social media page that is dedicated to someone that has fans, usually set up by the fans themselves. A fanbase is the following that a celebrity/sports player etc. has, and refers collectively to everyone who is a fan of him or her. So fansites are often created by the fanbase. Two examples of the difference from the BBC: http://news.bbc.co.uk/sport1/hi/football/eng_conf/9506134.stm - "Worcester City director Mike Davies says the club are not intending to raise ticket prices as they look to build their fanbase." http://news.bbc.co.uk/local/northampton/hi/tv_and_radio/newsid_8339000/8339677.stm - "A Northamptonshire woman is enjoying the success of over seven million visits to her Strictly Come Dancing fansite."