In your example, they're interchangeable, of course, but I can think of many things that I would call "amazing", because they amaze me, but without finding them particularly impressive - or even while finding them sad or lame. And then on the other hand, if someone performs a notable achievement, I'm likely to call it "impressive", but not necessarily "amazing" if I had high expectations of that person.

Of course, I'm not a native speaker and as such may be inclined to take things a little more literally, make a more conscious connection between "amazing" and "to amaze" / "to be amazed" than a native speaker would do.