Twitter and Facebook do, in fact, use different versions of OAuth. Twitter uses OAuth 1.0A, while Facebook uses OAuth 2.0. These versions have different implementations.
For Twitter, there are two options for accessing the APIs via OAuth, application-only auth and application-user auth.
For details on Twitter's OAuth, check out https://dev.twitter.com/docs/auth/oauth
For details on how to obtain access tokens for Twitter APIs: https://dev.twitter.com/docs/auth/obtaining-access-tokens