I'm considering degrees (in England) and I noticed that Mathematics and Mathematical Sciences are separate. What is the difference? Is it the case that one subject is a sub-subject of another? Would either have no advantages to studying over the other?
You should consult the specific programs you are interested in to find what they cover and how they differ at that particular institution, and the advantages of one versus the other depend on your interests and goals.
"Mathematical sciences" suggests a potentially broader focus of study to include math-adjacent subjects like statistics, mathematical approaches to computer science, actuarial science, etc.
The specific focus of a given program depends on that particular program; a particular program awarding a degree in Mathematics could include study in 'mathematical sciences' as well as more of a 'pure math' approach, or it could not, and vice versa: a program in 'mathematical sciences' could include mathematics itself alongside other related topics. A program in 'mathematical sciences' in one institution might include mathematical statistics; in another institution there might be a clear separation with another program specifically in statistics and no overlap.
In summary: you should consult the specific programs at institutions you are interested in, most likely starting from their website (for example, for Oxford's MSc in mathematical sciences).
Advantages of one versus the other depend on your interests and goals, and can't be answered here.