As someone in the comments have mentioned, Bitcoin is not competing with Paypal, but rather it's competing with fiat money. Although that statement isn't strictly correct, I think there is some truth to it, since Bitcoin can act both as a money and a money transfer service. If you see Bitcoin as money, which it is, then we need to develop a service like Paypal that can handle a huge number of transactions per seconds. In my opinion when the Bitcoin community can achieve to do so, that service will one of the Bitcoin killer apps.

One may also suggest that Bitcoin doesn't even compete with fiat, because fiat are currencies, whereas Bitcoin is money, so Bitcoin is actually competing with gold.