I use both, for completely different applications.
Generally speaking, Stripe is probably the easiest way to start accepting credit card payments on your site. Just about every major ecommerce plugin for WordPress includes support for Stripe. And it’s a breeze to set up, compared to Braintree.
Braintree is a division of PayPal, and while their fees are pretty much the same as those of Stripe, it’s much more complicated to begin using, in my personal experience. You won’t find as many integrations available for WordPress ecommerce plugins, though the most popular ones will likely support it.
I’ve also found that their regulatory requirements can be quite a bit more stringent. I had to pay a third-party service to create and maintain all the compliance documentation that Braintree required. While Stripe is equally concerned with regulatory compliance, it’s generally much more user—and developer—friendly.
I would recommend Stripe over Braintree, based on my personal experience. But there’s no easy answer as to which payment platform is better, or which one you should choose. Your choice should be based on your business needs, the payment methods your customers prefer, and whether or not there are integrations available for the ecommerce plugin(s) you’ve chosen to use on your site.
Hope this helps.