An online rival is copying us. Should I launch an ad campaign to point this out?
If your website offers a better user experience than your competitor then you should be okay.
Consider the following:
1. How well do people know my brand versus the competitor (if you're well-known and they are not, then ignore them).
2. Who has a better site experience (if they do, then ignore them).
3. Do you have other things you want to say about yourself (if so, are you better off saying these things?).
However, if you want to start a campaign against your competitor and you've considered the above, it could be a good thing to do.
Saying that, be careful taking this advice. I'm sure many other people would advise against it.
They'll mention things like you're giving your competitor free media, or that you are focusing on your competitor not on what you are doing.
However, here is the reason why I think you can do it – you'll grow the category.
Nelson Mandela (and I'm sure others) said you should always respect your enemy: "The stronger they are the stronger you are."
Hence engaging in open warfare with your competitor may raise attention to not only them, but yourself as well. If they bite back, then the category grows as well.
So go for it. But do it in a fun way. Do it in an intelligent, talkable manner.
Keep the tone fun and talkable. And weigh and consider all options before taking this advice seriously.