In my reading of French texts I have seen both
‹‹ J'ai mieux fait ››
and
‹‹ J'ai fait mieux ››
Is there a difference of meaning, nuance or usage between the two?
J'ai mieux fait is usually followed by que or de. In the first case, it is similar to J'ai fait mieux que [...]. In the second case, it usually means that you had a choice and you selected the best option (e.g. J'ai mieux fait de venir en Belgique que d'entrer en prison, Proudhon, 1861).
J'ai fait mieux is self-sufficient, i.e. can be a full sentence by itself. It means that you did something better than someone else (or than yourself in the past).
Yes, there is a difference sometimes :
Homever, both can be said like the first use I described, you can say both "J'ai mieux fait quelque chose que quelqu'un" or "J'ai fait quelque chose mieux que quelqu'un" which have the exact same meaning.