Is there a difference between забывать/забыть and позабывать/позабыть?
1a) Я забыл ее имя.
1b) Я позабыл ее имя.
2a) Слушай, на время время забудь.
2b) Слушай, на время время позабудь.
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In terms of style позабыть sounds a bit archaic or bookish. The one who chooses it over the neutral забыть creates the impression of being well-read/old-fashioned/highbrow.
In terms of semantics...
A verb with the prefix по- acquires the meaning of a distributed action: побросать - to have dropped many objects (independently), поснимать - to have taken many objects off or down, понастроить - to have built many constructions etc.
Thus, позабывать means to have forgotten many things. However, позабыть (as in: Я всё позабыл) has all but lost this shade of "distributedness" and become almost synonymous to забыть. Hence the aforementioned archaic flavour.
The answer by Avi Gordon is wrong, позабыть does not mean forget multiple things. This verb does not have such connotation.
The semantic difference between позабыть and забыть is in that позабыть means to forget permanently, while забыть means that one cannot remember the thing now even with effort but may recall later.
Забыть also can mean forgetting doing something or taking something.
In short: позабыть is permanently, забыть is temporary.
Я забыл ее имя. = I cannot remember her name. The reason may be any, my bad memory, untypical name, I heard the name only in passing, long time gone, I parted with her etc.
Я позабыл ее имя. = I erased her name from my memory (means I parted with her for good).