I have owned both (Still do) I have a 96 Civic LX and a 2001 GSR. Personally, I don't think either one is better than the other. They both have their benefits. Both cars are built on the same plattform, but the Acura is more of a luxury car then the Civic.

Here are my thoughts on both:

Civic -
Easier to haul friends in the four door
Better gas milage
More roomy
Lower insurance
Easier to work in engine bay
More aftermarket parts availability at a lower price

Integra-
More power
Handles better (susp. & brakes)
More of a luxury feel
Most aftermarket parts you would have installed on a Civic are already there

Like everyone else said, it all depends on what you want to use it for.