Am I allowed to say on my site that my company is better than someone else's company?

Only if you have facts that back up your claim. While some bragging is allowed in advertising, various government agencies, not to mention the other company's attorney, will come knocking on your door if you stretch the truth too far.

With respect to what? Is this something you can prove IE if your company delivers all products within 2 days and their company delivers all products in more than 3 days? Don't say anything that you can't back up - with hard evidence.