Bitcoin is a cryptocurrency, a digital decentralized payment system working without any central repository or single administrator. All transactions in this are through cryptocurrencies and unspent amounts are counted in Satoshis. While paypal is an online payment system to facilitate transactions in US$ only. One must have a credit card to use Paypal, while its not applicable in Bitcoin case.
Bitcoin has lower fees than Paypal. Paypal is not available in every country, but Bitcoin is border-agnostic like the Internet.
In my view, Paypal and Bitcoin have no comparison and hence, no question to overcome each other at least in the current scenario.