Mutual Authentication & Oauth are two different authentication mechanistic.
Mutual Authentication: Allows client and server to identify and authenticate each other by using certificates.
Ref: http://www.aboveandbeyondcloud.com/mutual-authentification-salesforce/
OAuth: also identify and authenticate user but using user/pass on salesforce site, so user do not need to reveal his password to 3rd part application.
Ref: https://developer.salesforce.com/page/Digging_Deeper_into_OAuth_2.0_on_Force.com
And yes, once the authentication is done we can consume any API (REST, SOAP, Bulk ..etc)