The ultimate difference isn't necessarily down to remote vs local. You can run a "local" model on the cloud the same way you could on a local machine and have all the same control over it. You can modify the weights, inject behavior, stack on arbitrary LoRAs, use whatever runtime you want, etc.

The problem arises in the case of anthropic, openai, etc. where you can't do those things. I think they have LoRA services that they'll charge you for, but that's about the extent of it. Think of it like having a compiler and its source code, and the operating system it runs on vs simply having access to invoking it via a web service.