我应该申请新H1b还是transfer H1b?

My ststus: I currently work as a postdoc for University A and hold an H1b visa which will be expired soon on Aug, 2004. Now I got an offer from University B.

My question: Should B do an H1b transfer or apply a new H1b for me? I don't know what's the difference. And is it too late or not if I just get the receipt notice (not final approval) from USCIS before Aug 2004?

Thank you very much for your kind help!