I think learning from the mistakes of others is a smart thing to do. That being said, if you're confident that you know what your customers want then you should release it as soon as possible. Also, the sooner you can get feedback from customers, the sooner you can make improvements. If you wait for your competitor to release, you might be making your tweaks at the same time the competitor who released is making their tweaks but they're making tweaks based on feedback from their customers and you're guessing.
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You need to pay attention to the competition to see where you stand, and how you can differentiate your app from theirs, but don't lose sleep over it.
You should be the leaders and not the followers (IE: they are copying your good features rather than the other way around).
Would it be smarter to wait and see what they have, in case we need to tweak stuff (to differentiate)?
I think learning from the mistakes of others is a smart thing to do. That being said, if you're confident that you know what your customers want then you should release it as soon as possible. Also, the sooner you can get feedback from customers, the sooner you can make improvements. If you wait for your competitor to release, you might be making your tweaks at the same time the competitor who released is making their tweaks but they're making tweaks based on feedback from their customers and you're guessing.