Google Wallet vs PayPal

Google Wallet
Google Wallet is an easier way to pay in stores, pay your friends and pay online. Shop in stores with all your loyalty, offers and gift cards in one place. Send money to friends and they can spend it instantly with the Google Wallet Card. Store your gift cards and loyalty programs and leave the plastic cards behind. Send money to any friend in the US with an email address or request money from them while on-the-go. It's easy, fast, and free to send directly from your bank account or Wallet Balance.
PayPal is an international e-commerce business allowing payments and money transfers to be made through the Internet. Online money transfers serve as electronic alternatives to paying with traditional paper methods, such as cheques and money orders. PayPal is the faster, safer way to send money, make an online payment, receive money or set up a merchant account.
Face to face in the news:

2015 - Google acquired Softcard to power its Wallet to catch up with PayPal

Just after Samsung acquired LoopPay to challenge Apple Pay's increasing dominance in mobile payments, Google also makes its step. It announced that it has acquired technology from mobile payments app Softcard, and that Google Wallet will soon come pre-installed on Android phones from three of the four largest U.S. carriers later this year. Softcard app (like Google Wallet and Apple Pay) relies on near-field communications (NFC) for its mobile payments.  Google Wallet allows users to make purchases at brick-and-mortar retailers by tapping their phones on contactless payment terminals. Though it predates Apple Pay by a number of years, Google's NFC payments system never achieved the visibility of Apple Pay, despite positive reviews.

2013 - Google integrates Wallet into GMail to challenge PayPal

Google Wallet was launched a couple of years ago as a futuristic payment service for offline stores via smartphone. However, this trick had no success (even in the US) due to the lack of NFC-smartphones among users and the lack of NFC-terminals in the stores. But not to loose the beautiful name, Google first combined Google Wallet with its online payment service Google Checkout, and now Checkout will disappear and Wallet will remain the only Google's payment service. Maybe it will be a surprise to you, but if you have bought something in the Play Market (app or book), then you already have a Google Wallet account. And soon, you probably will be able to send money to other Wallet users and receive money from them (like in PayPal). And you'll be able to do it simply by sending an email from GMail and attaching the desired amount (as shown in the video). For while this feature will be only available in the US.