What's the difference between saying;
Will you be going home this summer?
Will you go home this summer?
Are there any differences between these in written or spoken English?
They are close in meaning, but one might be preferred over the other in some cases. Will can express intentions formed at the moment of speaking, while be going to is often used to talk about intentions we already have.
So, the question Will you be going home this summer? might be put as a general, isolated inquiry. On the other hand, Will you go home this summer? might occur in circumstances in which some recent event might put such a journey in doubt.
I don't feel there is a difference in written versus spoken text for either variation you ask about or any that I discuss below. I will say that the verb "go" is a very special verb and the use of these tenses with the verb "go" is somewhat different than the use of these verb tenses with ordinary verbs. If you want an explanation that applies to the verb tenses in general, please say so and I will give you other examples.
I disagree with SomeNorCalGuy about Will you go home this summer? necessarily being curt. It can be curt, but it can also be neutral or pleasant.
is very neutral and inquisitive. On the other hand, Will you please go home this summer? is asking if you will comply with my request that you go home. Because of that, Will you go home this summer? can sound like a request or command that is ruder because you did not even say "please."
Will you be going home this summer? is more common in British than in American English, and because Will you please be going home this summer? is improper grammar, Will you be going home this summer? does not run the risk of being misinterpreted as a request or command.
As a native American English speaker, it is almost impossible for me to read Will you be going home this summer? without hearing it in a British accent. So while I expect it is common usage in England, in the U.S. I would consider it excessively formal. Are you going home this summer would be the most common way (in American English) to ask without implying a request.
They mean essentially the same thing: Are you going home this summer? which is probably the most neutral form of this question.
However Will you go home this summer? might be seen as being curt and could imply that you wish for that person to leave. It could essentially be taken as Will you please go home this summer? but without the please.
Will you be going home this summer? is more inquisitive. It implies that you are merely curious about to their summer plans and you don't particularly care what the answer is.
If this is being addressed to a person that you do not wish to go home for summer you could try asking You aren't going home for summer, are you? which retains the overall neutral value but implies that you prefer for them not to leave.