What is the difference between the following:
It depends on how soon I get my visa.
It depends on how soon I will get my visa.
"Will" implies something is going to happen in the future (in this case, getting your visa). But it's unnecessary in the first sentence: "how soon" provides enough context that you don't need the word "will." Like rudra, I'd interpret both sentences as conveying the same meaning. In fact, I might also suggest:
as that would essentially mean the same thing, too.
However, if you were asking the question:
then the "will" is necessary. (You wouldn't say, "How soon I get my visa?")
As Andrew Leach said in a comment, it is not usual in English to use "will" as an auxiliary in an embedded temporal or concessive clause. You usually see this with "when" and "if", but this is also a kind of temporal clause.