Started to use Stripe recently for a project and have used Braintree extensively for work. Your comparison is spot on.

There are some small differences such as, Stripe can send credit card details through AJAX to their servers so they are never sent to yours. Whereas Braintree uses Javascript encryption to do it, so the encrypted values are sent to your server. But I haven't seen anything that only Braintree supports yet.

Tradeoff with Stripe is that while you get a much simpler API, it (likely) will cost you more than Braintree depending on scale and what cards are commonly used.