A larger competitor recently sent us a legal letter asking us to change certain things about our app because it infringes on their app (it doesn't).
Is it legal for us to make this letter public on the internet and contact sites like TechCrunch in order to potentially get some publicity out of this?
Whether it's legal or not, I couldn't say.
It is probably OK, but that depends on the laws of the jurisdictions you're in. California is probably popular enough with the startup crowd that somebody here might be able to give you a more definitive answer in that aspect.
Considering that, at least in your opinion, these guys seem to be overzealous in legal action, they may come after you for defamation of character or the like as well. Are you really up for a second legal battle?
But beyond whether you could or not, I think you need to stop and ask yourself whether or not you should. Publicity is good, but you have to ask yourself the question of whether getting attention for the equivalent of, "Hey everybody, these guys are picking on me!" is really what you want.
Some people will see you as being whiny or entitled. (Especially the people who are loyal customers of your competitor.) More importantly, it would seem to me that in the best case scenario for you, people's opinion of your competitor will drop, but that doesn't necessarily translate into their opinion of your company rising.
I'd say you're better off spending your efforts and energy building a better product and taking that to TechCrunch instead. You may not get very many (if any) opportunities to convince sites like TechCrunch to listen to you. I'd hate to squander the opportunity by using it to try to tear down another company, instead of selling them and their visitors on your own.