There is a difference in that 'minnas' is something you do effortlessly, of events which you really would not forget easily (such as your childhood or your wedding), whereas 'komma ihåg' designates an action where you either put some effort to remember something, such as locking the door when you leave home so you wouldn't have to worry about it, or it refers to an action where you are asked by someone of an event that took place, or of something you didn't pay much attention to at the time and now when asked you have to rattle your brain to recall it. In the gray area between these extremes the words are interchangeable. Note also that the words negate differently:

'Jag minns min barndom' - 'Jag minns inte min barndom' or even 'Jag kan inte minnas min barndom' 'Jag kom ihåg att låsa dörren' - 'jag glömde låsa dörren' (this is the first case, see above) 'Jag kommer ihåg att Jimmy var med' - 'Jag kommer inte ihåg om\huruvida Jimmy var med' (the second, see above) or 'Jag kan inte komma ihåg...'

Also while I'm at it, if there's a sudden change in your knowledge about something use 'komma ihåg-glömma' as in

'Jag har totalt glömt bort min barndom' (because of a blow to the head or something) 'Jag kommer nu ihåg att jag visst låste dörren' (sudden recollection of a fact)

This was a bit sketchy though, but I'm sure someone will come along and correct it if necessary. Good luck!