The basic meaning is the same, but they come from two different histories. "Half" has a Germanic history, whereas "semi" has a Latin background. We gained almost all our Latin-borrowings through French-speakers. Sooooo... the result is that certain words tend to match up well with other words from the same background: halfway, halftime, and halfwit are all Anglo-Germanic. The words 'desert' and 'final' come to us via French/Latin, so "semi" fits them better. This would be the good rule to apply. ;)

Half isn't combined with other words as much as semi is. When it comes to words like semifinal, you just have to learn that word. This will be a difficult word for you to learn because there is no simple answer. Semi sounds odd in places and half sounds odd in other places. It's just based on common use. There isn't a good rule to apply. Semi- is a prefix. It is attached to the word. Example "semicircle". Half is usually separate. Example "half circle". Semicircle and half circle are interchangeable. Half final is odd and most English speakers wouldn't understand. If I was you, I would consider them equivalent, but learn to match your use of them to what English speakers normally do. Sometimes you'll have to change a sentence based on which you use. Here is an example: I go to the movies every half month. I go to the movies semimonthly.

They are both right in their answers but from what I've learned in English this is how you tell the difference between to two. Half is right down the middle of a circle meaning two equal parts. Half-time is the middle of the game. Half-way is the middle of where you are going. Half-wit meaning you only have half of your wits. Semi means more like "almost". For example: Semi-circle means it is almost a circle semi-final means it is almost a final semi-bold means it is almost bold semi-colon means it's almost a colon semi-desert is almost a desert There are exceptions (obviously) and they are interchangable like semi-circle/half circle because semi can also be half and half can be semi but over all I think this little trick will help you.

you may check the history of english word