I'm curious about the general difference between the two, but I have to be more specific.
I'd like to know the difference in the context of the noun period — should I use "the start of the period" or "the beginning of the period"?
Both mean pretty much the same thing, but start is felt to be slightly less formal than begin (possibly a result of the extra syllable in begin).
Both words occur in most idioms that feature one or the other (these sentences are all grammatical and also synonymous):
But the second sentence in the following is ungrammatical, so there's some difference, at least in the verbal usages:
The main difference is in the etymology.
'Start' comes from Old High German; 'begin' is probably West Germanic. We also have 'commence' that came from French, and hence has a connotation of being in a more educated register.
In use, the difference between 'start' and 'begin' is that 'start' has an idea of suddenness, as in 'startle'. 'Begin' implies a process, a sequence of events.
If you say "Shall we start?", it implies right now, that it is urgent, and the change from not working to working is important. If you say "Shall we begin?" is more relaxed, and implies that something has been organised, and the beginning has a schedule following it.
You could maybe think of it that a 'start' is a transition from one state to another, and a 'beginning' is the origin of a continued path, journey, or process.
On that basis, to answer your question, a period should probably 'begin' rather than 'start'.
The only differences I would loosely ascribe to the those two words in the given context are connotative:
Start may have the connotation of being in the future and beginning may more easily be associated with the past.
Start has the sense of being a fixed point in time, while beginning could possibly refer to any time between the start and the halfway point.
Above based solely on my own perception as a native speaker of American English with a west coast bias.