I think that there isn’t usually a difference, but in the example that you provided there is a difference. “Would you like to dance with me” is asking someone to dance. “Would you like dancing with me” is asking if dancing with you would be enjoyable.

'like' is usually followed by verb+ing (meaning 'enjoy') Example: I like skiing. (I enjoy skiing) You could, if you wanted to, follow 'like' with the full infinitive (to go, to sit, to be, etc.) Example: I like to ski. (However, when it expresses enjoyment, the -ing form is more common). There are times when you have to use one or the other. HOWEVER 'would like' is followed by the full infinitive (to sing, to have, etc.) 'would like' is not the same as 'like' - it expresses a hypothetical situation. As Audrey says, 'Would you like dancing with me?' means 'Would you hypothetically find dancing with me enjoyable'. It's very unlikely that anyone would ask this.