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The defender, PayPal, an US-based company established in 1998 with billions in revenue.
The challenger, Bitcoin, the world's first decentralized digital currency.
For most people using PayPal is an acceptably secure way to pay online. Importantly the service shields your financial details from the seller and they offer both Security Keys and MTAN. However, PayPal is a common target of phishing emails which can be very sophisticated and easy to fall prey to. If your account is compromised, it will likely be sold on the black market to the highest bidder and worse could leak your bank account or credit card details.
At its core Bitcoin promises to be the most secure Payment method available, there is no database to leak or accounts to be hacked. However, Bitcoin transfers a lot of the responsibility for Security into the hands of the User which can be dangerous for those who don't know what they are doing. A Bitcoin wallet holds all the information needed to make transactions from a particular account and is now a target for thieves and viruses. However, with the advent of encrypted Wallets and a new breed of online-wallets such as My Wallet it is now much easier for the average user to keep their wallet safe and secure.
PayPal has had years to refine its user interface and checkout procedure. Payments can be made instantly with any credit or debit card and requires no intermediary or exchanged. PayPal also has a chargeback policy, which favors Buyers over Sellers providing more protection for Users in event of problem with their purchase.
PayPal has a large advantage here.
The usability of bitcoin is severely hampered by the need to exchange the User's domestic currency into Bitcoins before a purchase. As Bitcoins do not support chargebacks this typically makes it difficult for exchanges to accept deposits by instant payment methods such as credit card or PayPal.
However Bitcoin has made improvements in other areas recently, the client is now much easier to use for the average user and with services like My Wallet you can manage your bitcoin's with an easy to use familiar interface.
PayPal provides a full range of Merchant API's and is supported by all major shopping cart software. However PayPal's chargeback policy can unfairly penalize merchants who sell digital goods or other virtual items. A plethora of horror stories are available from merchants who have had malicious chargebacks cripple their business or who have had their funds frozen by PayPal for no reason.Famously PayPal blocked donations to the whistleblowing site Wikileaks which made it difficult for them to fund their operations. Fortunately they were able to begin accepting bitcoin donations soon after.
Services like bit-pay make accepting bitcoin's as easy for merchants as accepting PayPal, funds can be immediately exchange for domestic currency so exposure to exchange rate fluctuations is minimal. The advantage for merchants is that as bitcoin is digital cash it does not support chargebacks, funds cannot be frozen and payments cannot be blocked.
Big win for Bitcoin.
PayPal accounts are tied directly to your bank account or credit card and PayPal is a regulated financial institution in many countries. PayPal payments are not in any way anonymous and it is not recommended you make purchase using PayPal that you would not be comfortable with the authorities knowing about.
A history of every bitcoin transaction ever made is available right here on this site. However transactions do not need to be tied to a bank account or individual and they are essentially anonymous if some basic precautions are taken. My Wallet can hold up to 1000 unique bitcoin addresses and it is recommended you change addresses regularly to avoid leaving a trail.