I enjoy both.
I enjoy stories about people who have powers wholeheartedly choosing to do the right thing despite all the opportunity they have to lead themselves in to utter debauchery.
I enjoy stories about people without powers who break their bones abiding by their principles, knowing that they could be crushed but never relenting.
I can't speak for Batman himself RECENTLY since every comic I read involving him or his family (besides Jason of course) usually ends up being a test of my patience (Metal has been pretty fun though), which then just devolves in to me PRAYING that now that Watchmen is getting hamfisted in to the mainline DC Universe at least Rorschach finally gets to start offing SOME of the sick, twisted bastards that the Bat-Family are too absolutely pathetic to finish off.
I mean would it ACTUALLY be so bad if some stuffed a grenade in Professor Pyg's throat and tossed him in to a dumpster? No, no it would not.
(No offense to you, Jumbo)
But I digress, this'll just end up being a whole different argument that has a very blatant correct answer outside of "Then what will they write about?", because the appeal for that type of character sort of begins from there.
The appeal for characters with powers such as Superman ticks for me, not because of the powers themselves, but because of the strength of his character and how he applies in increasingly bleak scenarios.
The idea that instead of all the wrong circumstances leading to a bleak and monstrous force of nature that wants to wreak havoc on an unforgivable earth, we have a being that found itself empowered beyond imagination compared to its peers. The difference comes in the fact that this being has been shown all the good that it could protect along with the redeeming aspects of the world that are worth preserving, most especially life. And so with its overwhelming strength in tow it chooses to defend life, without trying to control it, even when the world keeps pushing that care to the brink.
The appeal for characters without powers, or at the very least really crazy ones is the idea of an average person, easily susceptible to grabbing for the closest venue of power, choosing to act as they are, their bodies be damned. No matter how much pain they need to endure, now matter how much their bones might ache for an end, they push on for the sake of everyone but themselves.
Both of these can hit close to home in regards to how their humanity comes in to play and how easy it would be for them to fall in to slumps, knowing what they'd risk for it.
In the end though it is just personal opinion, but I hope my answer contributed to the discussion.