a) Przedstawienie będzie grane do końca stycznia.
b) Przedstawienie zagram do końca stycznia.
first of all this is not a good example to show how future perfective differs from future imperfective and it's because 'będzie grane'
is first of all passive voice - that's why the translation I given was 'will be played
and if you form a perfective future of passive voice you use a construction using the verb 'zostać' - in this case 'Przedstawienie zostanie zagrane do końca stycznia'.
that's why your second sentence (b) is not correct in this case - as 'zagram' is active not passive voice - these two sentences are not counterparts
For me, a) states: The show will be played until the end of January. But some uncertainty remains about the end of the show (maybe it will be shown a week shorter or a week longer).
The show will be played until the end of January (repeatedly or continuously).
The correct grammar perfective counterpart 'Przedstawienie zostanie zagrane do końca stycznia' means 'The show will (be ready and) be played once before the end of January. - perfective actions are one time or finished actions- in this case it is a one time action
to show you the active voice counterparts:
Zagram przedstawienie do końca stycznia. - Before the end of January I will have played the show.
Będę grał przedstawienie do końca stycznia. - I will be playing a show(the show - depending on context) untill the end of January
the strange thing is perfective and imperfective in the future tense can for some verbs have almost the same meaning with just a slight nuansical difference: 'Będę grał w karty.' is almost identical to 'Zagram w karty.' the differnece only shows in some contexts in which one should be used and not the other.
go on if you still have questions