In my native tongue I have an equivalent for "fan" and an equivalent for "supporter" which is much more commonly used than "fan". These two words clearly explain the difference.
Supporter is all about sports. S/he loves tennis, and s/he would enjoy watching nearly every player (except for the least liked on a personal basis).
Fans are running after a player/team, and don't notice anything else. When they come to the tournament, their main goal is to make 1000000000000000000000 pics of the centre of their universe, get his autograph (at least 5 times), mob him every time he leaves the protected area, run around the stadium for hours to find his practice court, etc. They are interested in the other players only as opponents of their idol, or DC team partner.
You wouldn't belive me, but I saw lots of these psychos in the tournaments where they bought fairly expensive tickets, came to the tournament from some other country, wasted money on their airtickets and hotel, and ... they would watch ONE MATCH PER DAY. Played by their idol. Though in a slam you can watch 5-7 matches per day(incomplete, but at least you see a lot of players, and can assess their level which is very amusing in the forst few days of the tournament). And in the masters events five matches are guaranteed in the Centre Court, and you can grab another couple of them peeping in the outside courts.
I am a Rafa Nadal supporter, and I could describe a typical day in the stadium of a typical Rafa fan.