"My boy has a new coat."
Translation:Mon garçon a un nouveau manteau.
Either can be used depending on the meaning.
Both words mean "new", but in different situations. Nouveau relates to something new that has changed, whereas neuf relates to something that has been newly-made. Nouveau precedes the noun but neuf follows the noun it modifies.
So if we used neuf for this sentence, it would read as follows: Mon garçcon a un manteau neuf.
Nouveau goes before the noun; neuf goes after the noun. It's all about context. (It might help to think of neuf as meaning "brand new" rather than just "new".)
If you think about new in English, it can have distinctly different meanings based on context. The same goes for nouveau and neuf. Sometimes the two are interchangeable, and sometimes they aren't. You just have to remember that nouveau goes before the noun and neuf goes after the noun.
_karsh is correct.
In French when you have a collision of vowels where the last letter of a word is a vowel which is confronting a vowel as the first letter of the next word it is common to slip an L in there just to make it easier to speak.
I used a similar device from English when I wrote the previous sentence. I wrote an L instead of a L because that is just what we do in English. I slipped an N in there just for comfort.