If the play is scored as a passed ball it is an error. It should only be scored a passed ball/wild pitch if a runner advances, or on the third strike the batter reaches first base safely. The scoring is sometimes a judgment call. Some statisticians are more strict on calling it a passed ball or wild pitch.
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The rules for pitchers and catchers are a little different than for the other players because they are involved in every pitch of the game. A passed ball is considered as a separate statistic and, therefore, is not considered an error. This is similar to a pitcher who throws a wild pitch. The wild pitch is not considered an error because there is a separate statistic to account for wild pitches.