They are basically synonyms. The difference is in the details, as is true for English. For starters, they both have による as a base. So we can immediately assume that the ‘depending on’ meaning applies. From here, the only work we need to do is look at the difference (違う), and then decide for ourselves if it is significant, or superfluous.
In English, just like almost every other language, we add extra words to sentences based on many things. Politeness, the need/lack of need for context, the person we’re talking to, the point we want to emphasize, etc etc. In this case, let’s pretend that a receptionist says both of these.
Clerk - ‘The price differs depending on the room.’ (Sounds polite as more words/description is added)
Clerk - ‘The price depends on the room.’ (Potentially sounds a bit less polite, due to the elimination of any excess language.
Now the same conversation with a friend after asking them about room prices.
Friend - ‘Price will differ depending on the room.’ (Quite polite for a friend, might make you feel like they’re being standoffish)
Friend - ‘Depends on the room.’ (Sounds friendly and pretty much exactly what you would expect them to say.
To answer your question conclusively. Yes, they are the same. But context will play a large part in how natural something sounds in almost all cases. Both English and Japanese are languages where sentences tend to get longer as politeness increases. Use this as a (rough) rule when deciding what sounds better in any case.