In certain contexts, both words have the same meaning. When you ‘peek at something’ or ‘peep at something’ you take a look at it very quickly; this is often done secretly and quietly. It is also possible to peek/peep at someone.
*I would like to have a peep/peek at the new play that she is writing.
The word ‘peep’ is also used in certain informal expressions. For example, if you say that you don’t want to ‘hear a peep out of someone’, what you mean is that you don’t want the individual to say anything or make any noise.
*We became rather worried because we hadn’t heard a peep from them.