I can only see two reasons why someone might need PayPal rather than Stripe: if the business resides outside the US, or if they need to take methods of payment other than credit cards. Other than that, Stripe seems like a no-brainer.
1) Costs. We pay 2.2% + $0.30 USD per transaction vs. Stripes' 2.9% + $0.30 USD per transaction. At our transaction rate, that makes a difference in received revenue.
2) Moving. Refactoring a well established business to interface with a new payment provider can be a costly and expensive process. Think of switching personal banks.
3) Fraud. Paypal probably has the most advanced fraud protection in the online business today. If you deal in any decent amount of online payments you will find this to be a huge issue.
Additionally, we have our own dedicated Paypal account manager that actively reaches out to us and keeps in contact, and we can instantly call a direct line to this person if we need assistance with an issue.
With that said, our team is actively looking at Stripe because we like the ease of use when it comes to reoccurring payments, which we don't do at this time but plan to implement in the coming year. From what we see, reoccurring payments are a messy with Paypal and I'm not satisfied with Paypal's solution compared to their competitors like Recurly, Chargify, and, well, Stripe. But I am impressed with what Stripe has to offer on all fronts. Especially their API which appears to be very easy to use. ...And, they have a nice looking reoccurring payments process which fits our vision nicely.
Anyway, disclaimer: we run between 50K and 100K a month through Paypal, so that is probably why we get fantastic support from them and better pricing.
When you're doing hundreds of thousands of dollars (or more for you big guys) in credit card transactions, eating 1% just for a friendly API is a huge turnoff.
Please Stripe, prove me wrong and provide an external merchant account option!