You are correct that level 60 is the current maximum level. However, as of patch 1.04, any experience earned after reaching level 60 contributes to your Paragon Levels.
Paragon Levels (Source)
- There are 100 paragon levels that a character can earn
- Each paragon level increases a character's Strength, Dexterity, Intelligence, and Vitality permanently and adds a permanent 3% bonus to both Magic Find and Gold Find
- The amount of time it takes to gain the first Paragon Level is close to the time needed to increase in level from 59 to 60.
Another gameplay change that occurs at level 60 is Nephalem Valor. This doesn't permanently improve your character, but it does allow them to gain some bonuses to gold and magic find that last until they log out or change their abilities. See the link for a full description, but here's a summary:
Nephalem Valor is a buff that becomes active for level 60 players when killing an elite monster.
...When you kill some of the previously mentioned monsters, you get a stacking buff that increases both gold and magic find bonuses.
...if you change a skill, your Nephalem Valor bonus drops off.
Edit: As Holger pointed out, switching acts also causes you to lose the Nephalem Valor bonus.
Also, it isn't mentioned ion the site that I linked to, but there is a limit of five times that NV can stack. Killing more elites/champions will continue to refresh the bonus, but it can't stack more than five times.
The linked site also sounds uncertain as to whether changing equipment will reset the buff. I can confirm that changing equipment and/or potions does not reset the bonus.
Another benefit of Nephalem Valour is that each stack of it increases the amount of XP you gain from killing enemies by 15%. This improves the rate that characters gain Paragon Levels.