Is there any benefit to leveling Pokemon?

The game says they "become stronger", and as you can see in the Pokemon menu (where you can set them as Support) for each Pokemon there's a Power value over its Ability. Pokemon with higher level deal higher damage.

For example:

• Level 1 Bulbasaur: 40 power
• Level 2 Bulbasaur: 43 power
• Level 3 Bulbasaur: 46 power
• Level 4 Bulbasaur: 48 power
• Level 5 Bulbasaur: 50 power

...and so on.

It's worth noting that the max level your Pokemon can reach is 10.

Usually the final Power of a Pokemon is the Base Power increased by 20. (Only 30 Base Power Pokemon have their final Power increased by 25)

UPDATE for version 1.3:

With the newly added Raise Max Level Enhancement, some Pokemon can now reach level 15. According to Pastebin, here's the complete BP/level chart: