About celebrex: It is right that it is slightly better for the stomach (GI tract in general) and it does not have the detrimental side effects (increased cardiovascular morbidity and mortality) that vioxx and bextra had.
Mobic is a less specific COX-2 inhibitor and has never been shown to be any better than celebrex.
As mobic is not generic it is as expensive as celebrex.
If you are looking for a cheaper alternative to celebrex, I would recommend Etodolac as it is generic and overall better tolerated than ibuprofen or naprosyn.
From a personal standpoint: my mother tolerated celebrex ok but had severe stomach problems with mobic. As always YMMV.
Disclaimer: I am not a paid consultant for any of these companies and have no direct financial interest in them.

Celebrex is classified as a COX 2 inhibitor used as a pain reliever. Mobic inhibits both COX 1 and COX 2. Both are used for arthritis.
Simple explanation is that Mobic may be a bit stronger but more likely to cause stomach irritation. Less simple answer is whichever NSAID works best for you is your best choice. If the Celebrex is working ok, why change? If not, try another NSAID. There are many excellent choices available, assuming you don’t have ulcers or heart problems which contraindicate using any type NSAID (non steroidal anti inflammatory drug).

Celebrex is called a COX-2 type drug. There’s much less chance of stomach irritation than with other arthritis drugs.

I doubt seriously that Mobic is the only arthritis medicine your acquaintances hospital uses. What if someone is allergic to it?

What it comes down to is that your doctor knows you better than your friend. Question him/her at your next stay to answer your questions.