At first: DON'T power up Pokémon before Level 20. Reason being: before Level 20 the single most effective thing is Leveling up, and that is best achieved by evolving, not powering up.
Due to a huge spike at level 20 (25.000 xp for level 20, 20.000 xp for level 21, 75.000 for level 22) it starts paying off more if you actually Power Up your Pokémon.
And also, a Pokémon caught on a higher level not only has stronger starting stats but also tends to have a higher potential, at least in my experience.
You can actually calculate a Pokémon's IV (Individual Value, the higher it is the higher its potential) from a number of stats.
EDIT: since some people seem to want to know it, it is always possible to catch a Pokémon which can get the best possible attributes when maxed out, however it often doesn't pay off if you level up ones that are on a low level to begin with since they eat a lot of resources.