I have both brands and have have not had any problems with either. I lend more toward the Specialized these days. But it would be nice to know which ones you are looking at. I think you are looking at the Fuji Newset 2.0 and the Specialized Dolce. If you are female the Dolce would be a better bike for, it is setup for a female. The Newest will mostly have a little longer crank and the saddle will be for a male, which should be swapped out for the proper one.
I would suggest you ride both and then buy. Make sure you have your helment, shoes and shorts, if you have them. Also take a ride on the bike, not around the parking lot. If the shop want to sell you a bike a 20 min ride shouldn't be out of the question.
Hope this helps.