Asking how "warum" and "wieso" differ is like asking how "why", "wherefore", "how come" and "how is that..." differ. In English, the words have more or less the same meaning and it appears that, in common usage, "warum" and "wieso" also have more or less the same meaning. One could argue that "wieso" asks a deeper meaning as to how it came to be that a particular fact is true but at least one German (the one who wrote the Yahoo entry posted by memcmp_) believes that most Germans don't draw any distinction between them even if teachers of the German language and linguists might. The author of the Yahoo entry writes "Warum" is the all-purpose word for asking "why" while "wieso" asks for "the reason why". "Weshalb" is asking more specifically about the cause. And yet, the author points out that the terms are used pretty much interchangeably in casual conversation. If you can't read all the German in the Yahoo entry, try copying the text into Google Translate. (My German skills aren't quite good enough to understand the text easily so that's what I did.) Of course, GT butchers the translation but you can get the gist of what the author intended and it's worth the effort.

To me, there is no difference between wieso and warum. I use them interchangeably.

I'd say they mean pretty much the same. Take a look at this: https://answers.yahoo.com/question/index?qid=20070919145633AANhWMo