The vigil is the Mass of anticipation for the following day. Going to a vigil Mass on the eve of a Sunday or a Holy Day of Obligation satisfies the obligation.
There is no difference between a Saturday evening vigil and a Sunday Mass. The same readings are used. It is the same Mass. There are some feasts (Christmas and Easter are the only ones I can think of off hand) where the readings for the vigil will be different from the readings on the feast itself (and the readings on the feast itself might differ depending on what time of day you go). However, even in such cases, the attending the vigil fulfills the obligation.