Lutris is for managing games, and can use multiple different engines. Proton is one, but also Linux native games, dos, ScummVM, etc. Lutris also interfaces with popular stores like Steam, Epic, GOG etc. It’s a game and gaming library tool.

Bottles is a general purpose wrapper for Wine. You can run games but also any wine software. It’s a general purpose wine tool.

Lutris makes running games in proton easy. Bottles makes running apps in wine easy.