Kimmel Center area
We were just at Amis a week ago. It def has more of that special occassion feel without being too stuffy (i.e. rock music plays but at a civilized volume). And if you are vegetarian, there are very compelling pasta alternatives that won't break your bank. I've yet to visit Estia so can't weigh in between the two.
I also really like Giorgio's and Viola from a food & cost perspective. Giorgio's probably comes off a little bit less "special", though you're sure to get a visit from Giorgio, one of the most personable proprietor's I've come across in the city. And Viola is so loud and crowded that you end up not having quite such a special feel. Amis def. trends towards loud and crowded, but I think the overall attention to detail compensates. The couple next to us had a 5-7 min lag between the arrival of their coffee and the arrival of deserts. The couple said nothing, but the slip up was met with profuse apologies from both the waiter and manager and an extra desert (which seemed simultaneously generous and, since they had each ordered a desert already, excessive to the point of punishment)
Another idea is Giorgio's on Pine, less than two blocks from the venue. It's Italian, BYOB, reasonable, and good. But, like Amis and La Viola, gets quite crowded and loud, especially pre-theater. I think that Estia might be your best bet for a birthday dinner while staying on budget, it's a nicer space and more "special" feeling to me.
La Viola - 253 S 16th St, Philadelphia PA19102 39.947775 -75.167742
Almost everything is under $20, and as a BYOB, you can bring a nice bottle of wine with you and save on the bar tab. Just remember it's cash only.
Another very nice option is the pre-theatre pre-fixe at Estia, which is Greek Mediterranean, and fabulous. The pre- dinner deal is a $30 three course, which is very affordable! The space is lovely too.
+1 for Estia, which has some of the best fish in Philadelphia.
I am partial to Amis, Vetri's smaller restaurant. Five minute walk to the Kimmel Center, so its within easy distance if you park the car near the Kimmel.
Some might say Amis is not on the cheaper side of menus, but if you are judicious, and you are eating early, (and therefore a bit lighter), the anti pasti and the pasta dishes can be had for reasonable prices. Drinks at Amis are a bit pricey. But the flavors there, especially the house cured salumi, make it well worth it. If you go this route, reservations are a must. It gets busy there.