In any RPG, leveling up means that your character gains stronger. You gain additional stats, skills and be able to wear higher level of gear.
This is true in Diablo 3 from levels 1 through 60. You gain skills, stat points, runes and you are able to equip higher tier equipment as you progress through normal, nightmare and hell.
You can actually see your character get stronger.
Once you reach level 60, you are no longer gaining skills, runes or the ability to wear higher tier equipment simply because that's the end of the line for them. For a while, this was the end of the road in progression through character stats.
Enter Paragon system.
Now everyone has 100 more levels to grind. But what's the point?
1. Magic Find and Gold Find. Reaching paragon 100 will actually yield you max MF and GF. At first glance, this is a pretty nifty idea. The premise is once you gain adequate paragon levels, you can slowly remove the MF affix from your gear and replace them with a much more useful one. And by reaching max paragon level, you no longer need MF or GF.
Problem: No one really cares about MF or GF(maybe some do, whatever). RNG kind of defeats MF's purpose. From the game's description, MF increases each item's potency. The higher your MF, the bigger the chance of you getting a high-affix item. So basically, if you get a 6 affix rare, RNG would still play it's part and not give you the right affixes for that item to be viable. MF only increases your chances of a high-affix item.
2. Stats. 300 main stat points. 200 vit points. 100 each for the remaining stats. Sounds really nice. But then again, in order to get those stat points, you would need to grind hundreds, if not thousands of hours to reach paragon 100. Or you could just get those stats from gear (:
Besides the MF, GF and stat points, reaching a certain paragon level should give you more than just a measly 3% MF and GF and a little bit of stats. Here's an example:
Paragon 10: +7 yards pick up radius (gets you a base of 10 yards)
Paragon 20: Health Globes and Pots give you +10,000 Life
Paragon 30: +10% Movement Speed Increase (stacks over speed cap)
Paragon 40: +50 Resistance to All Elements
Paragon 50: +500 Armor
Paragon 60: +12% Life
Paragon 70: +100 Vitality
Paragon 80: +100 Main Stat
Paragon 90: +5% Damage
Paragon 100: +10% Damage against Elites
Have an NPC in town that would check your paragon level and add the adequate reward for you current level.