No difference, same meaning.

I would interpret "I wore a new shirt out" to mean "I wore out a new shirt". To "wear out" a shirt means to wear it so much that it becomes damaged. All clothes wear out eventually if you wear them often. I find that shoes wear out after about 5 years. "I went out with a new shirt on" means I left home and went somewhere wearing a new shirt.

TWO meanings! Be careful! The others have explained. Although, actually, it would be difficult to wear a NEW shirt out! (to wear out - to wear it so much that it becomes thin and damaged). Ha ha!

It might be the same as in "I wore a new shirt when I went out" or it might also mean that the shirt is worn, in other words, used, with holes or other wear and tear. So "I wore out that shirt" is different from "I wore that shirt when I went out". The context will help you decide.

As a response to various comments you have already had: A. "I went out with a new shirt on" = I went out, and, when I went out, I was wearing a new shirt --- but, depending on intonation, I could be emphasising: (i) the fact that the shirt I had was *new*, or (ii) the fact that what I did was *go out*, or (iii) both or neither of those things and/or something else (for example, the fact that it was *me* who went out wearing the new shirt, not someone else). B. "I wore out a new shirt" = I wore a new shirt (presumably many times or for a long time or in a tough situation) and, as a consequence, it got into a bad condition (slightly damaged or similar) C. "I wore a new shirt out" can have two meanings: (1) exactly the same as meaning B, above, with the same nuance and emphasis, or (2) basically the same as meaning A, above, but normally emphasising the fact of wearing the new shirt as per A(i), and not emphasising the fact of going out (though depending on intonation I could be emphasising the fact that it was me who went out wearing the new shirt). Worth noting though that the intonation of C(1) would normally leave "out" unstressed (unemphasised), whereas in C(2) the word "out" could be more likely to include some stress (vocal emphasis) on "out". I guess this is because if the meaning involves "going out" then the word "out" refers to where you went, which is a central theme in the sentence. However, for C(2), if I had already mentioned that I went out, there might be no stress on "out". So it all depends... But I am not an expert :-)