I heard of some master's degrees that are just M instead of MS or MA like Master of Mathematics or Master of Psychology. Apparently, this is due to it being non-thesis.
However, I met people who choose non-thesis in MS or MA programs that have a choice between thesis and non-thesis. Their IDs say MS or MA. Is this because the program has a choice for thesis?
In general the formal name of a degree is a very bad guide to the contents of the degree; the name of the degree is heavily influenced by local history and traditions (some extreme examples: a Master of Arts degree from Oxford is automatically awarded 21 terms after matriculation to anyone with an Oxford BA; the degree that qualifies one to practice medicine in the USA is a Doctor of Medicine, while in the UK it's typically a combined Bachelor of Medicine and Bachelor of Surgery; a Bachelor of Philosophy degree at Oxford is a two-year philosophy degree aimed at those with a first degree).
In the UK there are increasingly degrees called 'integrated masters' (or sometimes 'undergraduate masters') whose names fit the pattern you mention. These are generally four-year degrees, generally in fields like mathematics, science and engineering, which one enters without a previous degree and which aim to leave students in a position to embark on doctoral study. See this example from Manchester Metropolitan University. Individual degrees may or may not include a mandatory or optional thesis requirement.
Your question contains two slightly different issues; this is an answer only to the more superficial part, about the abbreviations. Hopefully it can help clear up the confusion this part is causing, and allow more focus on the more substantial issue, about the content of the degrees.
The degrees you describe as “just an M”, like MMath or MPhys are not usually thought of that way, hence the confusion in comments. MMath is short for “Master of Mathematics”, just like MA is short for “Masters of Arts”. Neither is “just an M” — they’re both an M of something. The difference in the abbreviations is because for historical reasons, Master of Arts and Master of Sciences are used for degrees in a wide range of subjects, and so are very common, and have shorter abbreviations; where as Masters of other subjects have to provide a bit more of the subject name in order to be intelligible.
So the other half of your question can be rephrased as: what typically are the differences between programmes called MA, and programmes called Masters in Subject — e.g. between an MA programme in a mathematics department, and an MMath programme?
