One major application of research in integral $p$-adic Hodge theory is in proving modularity results, e.g. for elliptic curves. Here one wants to understand liftings of global mod p Galois representations to $p$-adic representations which have some reasonable local properties at $p$ (e.g. being crystalline or semistable); and to do this it's important to have a good understanding of the corresponding local problem at $p$, which belongs squarely to integral $p$-adic Hodge theory.

E.g. have a look at this famous paper:

Breuil, Conrad, Diamond, Taylor, *On the modularity of elliptic curves over Q: wild 3-adic exercises*.

This was the paper that completed the proof of modularity of elliptic curves over $\mathbb{Q}$ (finishing the job begun by Wiles in his work on Fermat's last theorem). The key ingredient in the proof is Breuil's work on classifying p-divisible groups over p-adic integer rings via semilinear objects ("Breuil modules").

This is just one of many examples where progress in integral p-adic Hodge theory has been instrumental in studying global modularity problems. You might want to look at recent works of people like Tong Liu and David Savitt for more examples of this kind.