Which hand should I punch people with?

Heavies have the option to punch people with either their right or left hand.

Does it make any difference which hand you punch with as a heavy?

The Heavy's melee damage is uniform for both hands. It doesn't matter which hand you pick, as long as you hit someone with it. In fact, even the hitboxes are the same - it makes literally no difference.

That said, however, depending on your personal playstyle (going even as far as how you press the mouse buttons), you may prefer to use one side - or have a better hit rate with one side. As there is no direct gameplay impact of your choice, you can then freely use your favourite fist to hammer into other people.

In my experience, the mouse button you click does not matter. The right hand is only swung when there is a crit. You can hear the sound that a crit makes every time the right hand swings. If i have my hands backwards let me know, because I have had multiple times where I click with the right mouse button and his left hand swings.