As Robusto commented, there is 'virtually' no difference between the sentences, and so they are interchangeable in the context you have given.
A more nuanced answer would be that the two sentences have slightly different connotations.
It is one of the best movies of my childhood is the cleaner, clearer, purer sentence of the two - it refers solely to your childhood in the sense that no other part of your life is being considered.
When you say It is one of the best movies from my childhood ... it is almost as if you are putting emphasis on the word childhood. This would indicate that you are comparing this period of your life with other periods. You could follow this with (something like) '... but when you talk about my university years, then that's a whole different thing!'