What is the difference between the arthritis medicines mobic (meloxicam) and celebrex?

Depends. Celebrex, or celecoxib, las a lower risk of bleeding from the gut, but also is the only NSAID that does not block the effect of low dose Aspirin used to prevent heart disease or stroke. Meloxicam blocks aspirins effect, as do all other drugs like Naproxen (aleve), and Ibuprofen (motrin). Unfortunately, Celebrex remains on patent and is much more expensive. For most, Meloxicam is a good choice.
Safety. Both are anti-inflammatory, but Celebrex blocks a specific enzyme called cox-2 at any dose, while Mobic (meloxicam) is nonspecific above 7.5 mg/day. The extrac GI safety of blocking the specific enzyme is lost at doses of Mobic (meloxicam) above 7.5 mg.