I'm assuming that the objective in your question is to have a colloquial or conversational way to discuss the difference between speaking and singing dialects (which is what you say in your second paragraph).
Professional singers will often adopt a different dialect for singing than the one they speak with. This is discussed in the article Why Do British Singers Sound American?, for instance.
That article discusses at least one technical term that is used as a measure of dialect modification between singing an speaking. That term is rhoticity, which is a measure of how R's are pronounced under different circumstances.
But rhoticity is just one aspect, and there are probably a multiplicity of others. And all of them would fail to meet the objective of having a conversational expression of what this is all about.
In that case, you might consider the term vocal dialect. This is used in ornithology for describing the regional differences in the singing voices of birds of the same species. (ref)
Your simplest choice might be to use the terms speaking dialects and vocal dialects, which, when used together in this context, are almost self explanatory. Throw in a rhoticity and talk about the Beatles if you must, to get your point across.
Another word that comes to mind, and it's also very colloquial is articulation. When it comes to singing, this is a common word used by vocal critics. (I learned this from watching Simon Cowell, who, despite his quirks, has a very astute and technical ear.) Articulation sometimes changes when switching from speaking to singing. For a much more detailed review of the topic, you might have a look at this article: Articulatory Phonetics. If you want to get more technical when talking about it, just about every buzz word on the subject is covered in that article.