They mostly mean the same thing - if you use 'good', 'better' or 'best' you are suggesting that the rest of the year is: not as good, worse, or great but not the best as the 'best' time of year, respectively.
However, they can mean slightly different things, depending on the context. The best part' is pretty straightforward - it is the best, full stop. The use of 'good' or 'better' may indicate that the year is not really that great in and of itself but there are times where it is bearable, more tolerable, and thus 'good'/'better'.
I would say that 'nice' can also be used to mean the same thing as 'good'/'better', however, it may be used specifically with reference to surroundings or environment - for example, spring time may be considered to be the nicest time of the year (because of how nice the weather is).