"In winter grandmother used the balcony as a refrigerator."
Translation:Зимой бабушка использовала балкон как холодильник.
There's no difference at all. It means the same. Just another good way to say that in Russian. You can perfectly use both of them and translate "to use (something / someone) as..." as "использовать (что-то / кого-то) как..." or "использовать (что-то / кого-то) в качестве...". The second phrase is a little bit more "official", maybe.
ABBYY Lingvo Dictionary, Essential Russian <-> English, 2015:
В качестве (кого-либо/чего-либо): as, in one's capacity (of), by way (of), for, as
в качестве варианта — alternatively, by way of illustration, as an illustration
в качестве основы (для) — as a basis (for)
в качестве отправного пункта — as the starting point
в качестве примера — as a case in point
You could and I probably would. 'My grandmother' might sound a little odd, however, if used in conversation with other members of the family who know the lady in question. If you just say 'grandmother' you are using the name of your relationship as a substitute for her personal name. My impression is that Russian people do this a lot, British people less so. Stylistically, if I were introducing my grandmother to a conversation, I'd probably refer to her as 'my grandmother' but after that, I might just call her 'grandmother'. But I wouldn't worry too much about it!