I am confused about the difference between the two expression.
I will do everything but shopping
I will do anything but shopping
thanks in advance
These sentences are generalisations, statements about ongoing preferences, which makes picking out the difference between them a little complicated.
Let's change the sentence a tiny bit and add some context.
This morning there is a long list of things to be done.
One of the tasks is shopping.
In the changed sentences I'll talk about "the shopping", meaning the specific shopping tasks that need to be done today.
Bill: I will do everything but the shopping.
Charlie: I will do anything but the shopping.
Bill is offering to do all of the other tasks, but will not do the shopping.
Charlie is being less generous. He is offering to do any of the other tasks, but will not do the shopping. I might ask Charlie if he would be prepared to do two, three, or more of the other tasks, but I don't know if he is offering to do more than one.
Back to the original sentences.
Because those are generalisations, there is not a specific list of other things to be done and I can't think of any significant difference between them.