Ya don't want to use the cheap Elmer's glue that's typically used in elementary school for paper craft projects.
I used Elmer's wood glue for years 'n' years in my woodworking adventures.
Elmer's & Titebond are PVA adhesives with differing "additives" whose purpose is to distinguish themselves from a marketing standpoint.
Strickly from strength standpoint, and imho, there's little/no difference. Elmer's produces a clear-colored joint, which is good in lotsa woodworking projects ... not so important in bow-building. Additives in Titebond do produce a more water-resistant joint, which is why I use for bow-building. It is claimed that Titebond cures to near full strength more quickly than Elmer's.
For bow-building, I've always used Titebond.