As @Matt said it's not that significant because they can be used interchangeably (and are both used, I use our in prayer all the time when with other people).
I wanted to address your comment to Matt's answer.
Heavenly Father is used so often in a indirect manner because of the personal relationship LDS feel they have with God.
If I was talking to my sister about my earthly father, I wouldn't say "Our father, said foo and bar", I would however, say, "Father said foo and bar".
He is our father, we know that already so there is no need to refer to him as such again.
It is much the same with our Heavenly Father. We know He is our Heavenly Father, there is no other Heavenly Father, and He is the Heavenly Father of all people on earth, so it's felt that there isn't a need to refer to Him as 'our' Heavenly Father again.
It's interesting to note that Our and My are probably used more by the LDS in prayer then in indirect speech. So if what you say is true about protestant prayer the usage of these possessive words is reversed between the two groups.
I would conjecture that it's more common in prayer in the LDS culture because of how formal and reverent the LDS are towards prayer. The LDS believe that prayer is a dialog between them and God, but the word choice is more akin to a dialog though mail (instant mail but still mail). Where the use of salutations like "Our" or "My" would be more appropriate then normal speech.