To "authenticate" is to verify the credentials of a user; in normal circumstances checking that the password supplied for the given username is correct. Sometimes this also includes making sure the user is permitted to enter the particular part of the site which they requested to enter (some areas could be restricted to certain users, for example).
To "log in" is to enter a site using certain credentials. This word describes what the user does in order to be authenticated by the site; in normal circumstances entering the username and password, and clicking a button. To be logged in is to have already been authenticated by the site, and usually means you get different pages given to you (ones that are customised).