The YubiKey does support the OATH-event based OTPs as described above, but the Yubico OTP takes advantage of the fact users do not need to type in anything to make the OTPs longer and more secure.
Each Yubico OTP has a plain text public ID as the first 12 characters of the OTP. This is used to identify the YubiKey which generated the OTP without having to perform any OTP processing. The remaining 32 characters of the OTP are an AES-128 bit encrypted hash. This hash is made of the Private ID, a string known only to the YubiKey and Authentication server, to further validate the OTP.
In addition to the Private ID, the OTP also contains counters tracking how many times the YubiKey has been power cycled (usage) and how many OTP events it has preformed since it's last power on (session). These counter values are stored in the authentication server and checked for each OTP. If the usage counter is less than the value on the server, or if the session counter is less than or equal to the value on the server, the OTP is rejected as a replay attack.
This means the YubiKey OTP will not get out of sync with the validation server, as well as adding additional randomness to the OTPs generated.
The new U2F protocol removes the need to have a central authentication server - the authentication process is just between the U2F device and the authenticating service. For more details, you can see: https://developers.yubico.com/U2F/
The YubiKey's main strength is that users can configure it to meet their own needs; YubiKeys have 2 configurations slots which can be independently programmed for the Yubico OTP or OATH-event based modes, as well as Challenge-Response or a static password. Further, with a small app on the host machine, the YubiKey can provide OATH- time based codes, identical to Google Authenticator.
The newer YubiKey versions also support the new U2F Protocol used by Google.
The YubiKey is developed and owned by Yubico - Yubico worked with Google to develop the protocol which would become U2F for Google's internal use.
Each Yubico OTP has a plain text public ID as the first 12 characters of the OTP. This is used to identify the YubiKey which generated the OTP without having to perform any OTP processing. The remaining 32 characters of the OTP are an AES-128 bit encrypted hash. This hash is made of the Private ID, a string known only to the YubiKey and Authentication server, to further validate the OTP.
In addition to the Private ID, the OTP also contains counters tracking how many times the YubiKey has been power cycled (usage) and how many OTP events it has preformed since it's last power on (session). These counter values are stored in the authentication server and checked for each OTP. If the usage counter is less than the value on the server, or if the session counter is less than or equal to the value on the server, the OTP is rejected as a replay attack.
This means the YubiKey OTP will not get out of sync with the validation server, as well as adding additional randomness to the OTPs generated.