Start may have the connotation of being in the future and beginning may more easily be associated with the past. The period will start in 15 minutes. vs I can barely remember the beginning of the period. 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.
The beginning of the century is a period of time which is short compared to the century but rather long otherwise; Some people may use this phrase to mean the first decade or even longer.
How to end a letter beginning with "To whom it may concern"