The meanings are very similar, and these three prepositions can be used almost interchangeably, particularly in the context of your "The mountain is covered with/in/by snow" example. But some subtle nuances may apply.
When referring to a substance that sticks to another, use in or with, but not by:
- The actress was covered in blood, or
- The actress was covered with blood, but not
- The actress was covered by blood.
- The ribs were covered with sauce, or
- The ribs were covered in sauce, but not
- The ribs were covered by sauce.
When referring something that physically protects something else, use with or by, but not in:
- The field was covered with a tarp, or
- The field was covered by a tarp, but not
- The field was covered in a tarp.
Use covered with to indicate an unusual amount of something on top of something else; use covered by to connote a covering so dense that the object being covered is completely obscured from view:
- The mountain was covered with fog.
- The mountain was covered by fog.
- Our grass was covered with butterflies.
- Our grass was covered by butterflies.
Somehow, the latter (covered by) paints a picture where the butterflies are so close together that I can hardly any the grass at all, but in the former (covered with), I picture a lot of butterflies, just not necessarily so many that I can't see the grass.
When talking about metaphorical coverage, use covered by:
- The roof damage was covered by insurance, but not
- The roof damage was covered with insurance, or
- The roof damage was covered in insurance.
- The city council meeting was covered by the news station, but not
- The city council meeting was covered with the news station, or
- The city council meeting was covered in the news station.
Other guidelines are likely to apply as well. This is not a question with an easy and straightforward answer.