There is a difference. The difference lies in the usage of those two expressions. Ich habe...gern is more commonly used with people. Like "Ich habe dich gern/Ich habe sie gern" and so forth whereas "Ich mag" is more widely used with all kinds of different things. Objects, persons, animals, just everything. You can say "Ich mag sie/Ich mag Käse/Ich mag meinen Hund/Ich mag es, zu schwimmen" and a bunch of other things. So both mean that you like something, but the usage as well as the grammatical background is different.

There's practically no difference in meaning.

I remember hearing, " Ich hätte gern Eis", oder "Ich möchte Eis bitte", "Ich mag Linenhemd lieber als Baumwolle". I am unable to define any usable difference between the two; so I'd just stick to the usual manner of usage and the best rhyme for my sentence. I'm far from proficient though in german.

I don't think there is any difference between them. But maybe others have a different opinion to this :) "Ich mag dich" and "Ich habe dich gern" seems the same to me. I use "habe gern" usually only for people but I am not sure if it is a rule or something :) for example: "Ich habe tanzen gern" sounds weird to me