I also always do the first run (for each character) in the default outfits, but I used the classic costumes for both of my second runs.
I think I like Claire's default better. I always liked the original costume in the sense that it introduced a burst of color into the RE universe at the time (at that point all playable characters had worn drab greens, grays, and blues) and was a good comic book/cartoon design. It also looked great on the original Toy Biz action figure. The problem is, even at the time, it looked dated. I kind of thought it was a design choice to dress her in something more appropriate to 1986 than 1998 in order to indicate that she was a bit backward and not a hip, city girl. But now it just looks ridiculously dated. I also didn't like that they tweaked the outfit to include a tank top rather than full tee shirt underneath the vest. If you're going to go retro, just go all the way.
I like Leon's alternate/retro outfit more. Probably just nostalgia talking, but I like the richer blue and enjoy the design elements of the bigger shoulder pads and white trim. To me, none of it is over the top enough to make it look ridiculous, but it adds a touch of color and flare to the look. I always thought the original outfit was a great design, and it definitely held up better for me in the remake. The default costume is a nice update, though, and I'm fine with it.
I've never really been into the more over the top secondary costumes in any of the games, so I doubt I'd ever play through with any of the goofier stuff offered here. The Elza outfit is a cool touch, though.