Just my two cents on this. The TV matches were all played in the 70's. I think there were three total. All were big hits on network television, to this day the most watched pool shows ever. Mosconi did win. He was the better player, especially playing games like 9-Ball or Eight Ball.
Saying all that, I agree with Huckster in evaluating both players respective abilities. Fats would be the favorite playing One Pocket or Banks against Mosconi when both were in their prime. He was definitely no "banger" as someone put it. The man could play, one speed below the top players in the above games. He played a sporty game of Three Cushions as well.
As far as gambling goes, it is NO match. Fats outhustled the world for many, many years. Even after he became a legend, he still matched up occasionally with top players (getting weight) and took it off. He robbed Richie Florence in his prime playing One Pocket for 300 a game in Johnston City. Fats was getting 8-7. He beat Richie for over 20K total over a two week period. Fats was one of the greatest pool hustlers who ever lived, but a total sucker at cards.
And now to put another myth to rest. Mosconi, for all his talent, was no saint or even close. He had a bad attitude most of the time, and was quick to scold people who moved or talked while he was shooting. Even if they were almost out of earshot. He demanded complete attentiveness at his exhibitions. Light a cigarette when he was shooting and you would get reprimanded. If he was struggling on a table, he would blame the equipment. Bad lighting, bad cloth, bad balls, bad rails, etc etc. It was never him.
And he was not always so gracious with his fans either, being caustic and brusque if he didn't like a question put to him. Heaven forbid, his opponent in an exhibition would play well against him. To Mosconi, this was a sacrilege. HE was the PLAYER, not this interloper. They were supposed to just roll over and let him win. Everything revolved around him in his world. Everyone else was just a spectator.
Fats loved everyone. He loved to talk to strangers everywhere he went, and would talk all day, enjoying the reaction he got from people. He was patient with people and would answer any question. He was not too self centered to compliment a child on his intelligence or a woman on her good looks. It was almost impossible to make him mad, unless someone asked him if he could beat Mosconi. He would put on exhibitions and make two or three shots total. He just talked for an hour or two and the crowd loved it. He was that funny. Fats WAS the Pied Piper, a people magnet. He could and did draw a crowd everywhere he went.
He also took care of all the stray dogs and cats in the town where he lived (Dowell, IL). He feed them every day and provided them with a secure and comfortable shelter. He was the ASPCA in that area. And he was always a soft touch for a pool player down on his luck, handing out 100 dollar bills to guys he barely knew. Thats the Fats I knew, and I'll never forget him. Yes, he was a hustler. But the man had heart, miles of it.