Prepare provides the sense that you are doing things now--making changes, getting ready--in order to be ready for something that is coming. Preparing may mean taking actions now that might prepare you for several different possibilities, so it may be a bit more general. Often but not always it is used when the thing to come is negative.
Plan means that we are discussing, thinking about, and deciding about something in the future. It does not suggest that we have started taking action yet. A plan tends to be a bit more detailed and concrete. Often but not always it is used when the thing to come is positive.
They are used about equally frequently, but they mean different things.