This distinction, as explained, is very tricky for me. I don't grasp the difference in meaning. Oh well....
la moitié de qqc -- half of something. La moitié is always a noun.Demi can be either a noun or an adjective, depending on how you use it.
une demi-heure -- a half-hour (here it is used as a prefix, which makes it a noun)une heure et demie -- an hour and a half. (here it's used as an adjective for the implied noun "heure).
Demi is usually used as a prefix with an easily recognizable unit like une demi-bouteille, une demi-heure, un demi-litre, etc. When used as such, it is always demi, never demie. But when you use it as "et demi", its gender is matched with the noun: une heure et demie, un litre et demi.
If you could specify where you are unclear on this particular lesson I'm sure that one of us will try to help you.
Also have a look here where we dealt with a similar issue:-
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