does a Mic using a tiny battery hit a better 'sweet spot' than...?
 

does a Mic using a tiny battery hit a better 'sweet spot' than...?
a Mic that doesn't use a battery?

Well, it all depends on the mic.

It's not really about where the power comes from, but how the rest of the mic is designed. Take any 2 different phantom-powered condensor mics, and they won't sound the same.

It is better to ask does this specific mic hit a better "sweet spot" than this other specif mic, or comperable non-battery powered mics.

Also, when you say "doesn't use a battery" it all depends... Do you mean a condensor that uses phantom power, or a dynamic that needs no power at all? For video work, I would stay away from dynamic mics, unless you are talking about having someone in the frame hold one in their hands. They are not meant for far away micing. They are meant for close up micing and are built to withstand high pressure levels (such as micing a screaming singer, a drum kit, or a guitar amp). For video work, you definitely want something like a small diaphragm condensor (boom mics are small diaphragm condensors, specially designed).

Alex F

Some microphones have an option where they can either be phantom or battery powered. They usually have slightly better specs while running off phantom power.

You probably want to be running phantom power where you can so you don't have to worry about batteries running out or leaking. But if you can't run off phantom, it's not a big deal to babysit batteries.