So your friend completed Legendary, got the achievements.
And it did not show that you completed Legendary on his card - (memory card I'm assuming)
And it says you had completed it on normal.
Firstly, Your friend has completed every single level on Legendary.
You, however, may have completed almost all the levels on Legendary, but maybe one or two levels you may have missed or completed on a lower difficulty setting. I suggest going throught the campaign menu and making sure all levels have been completed with the legendary symbol.
The reason I say this is because if you finish the game by only playing Legendary levels, you will still get the normal achievement. But if you miss one level and do it on normal, and all the others on Legendary, you will still only get the normal achievement.
Finally: The first intro cutscene level may count towards the achievement. Check if you have done it in Legendary.
Hope this helps
Edit: You don't actually have to watch the cutscenes, but the very first 'cutscene' also counts as a level, in a weird way, it happened for me too like that.