After level 18 there are 2 more missions.I will tell you the story with spoilers so if don't want to read, feel free to
Story Spoilers
After you beat the bosses, a cut scene shows up, in which Arkham absorbs all the power of the Force Edge and turns into Sparda. (In case you haven't played DMC1 Force Edge is that sword that Arkham found on a previous cut scene. After that Mission 18 ends.
Mission 19.
It too starts with a cut scene. It is in the library where Dante defeated Lady. Lady is still there sitting in a depressed mode, but there is also a certain someone walking. Lady sees him but doesn't do anything. The certain someone opens a door and disappears.
Dante is now still traveling by small portals who look like his amulet. The 1st one led him to a blood lake full with Abyss. The 2nd in a place with Mirrors which you need to smash in order to defeat some other Abyss. The 3rd in a place with an Hourglass which limits your time of exterminating the final Abyss guys.
After that Dante got the Samsara and went back to use it in a statue in the place with the Mirrors. After that the portal leads him to where Arkham is.
A small talk between Dante and Arkham starts, Dante mock Arkham's demon appearance so Arkham lets all the devil power overflow his body and turned in a big fat purple blob.
The fight starts but Dante can't defeat Arkham. Then, when he is about to receive a blow, Arkham's hand got cut off. Vergil has returned! The two brothers join the battle together and kick Arkham out of the Demon World with one powerful strike. Then a hole leading to a waterfall opens, the two amulets and the Force Edge fall in there and the brothers follow behind. Before reaching the depth each of the twins gets his own amulet and the Mission ends.
Mission 20.
In a cut scene Lady has reached the peak of the tower and is near the gate of the Demon World when Arkham suddenly falls off. Lady finally killed him for good, but still wept some tears in the end.After that the Mission starts.
Vergil and Dante reach the depth and Vergil gets a hold of Force Edge. Vergil wants Dante's amulet to receive ultimate power and Dante wants Vergil to stop all this so the begin the fight. After you beat the boss, a cut scene shows with Vergil staying in the Demon World and Dante leaving with his amulet and the Force Edge for he human world. He meets Lady and hands over the Kaleena Ann. They talk a bit, and Dante cracks a tear, showing that even a Devil May Cry.
They fight together the last demons of the game. If you beat more than 100 demons you get a Vergil cut scene in the end.
After that, you see Dante in DMC1 attire having named his shop Devil May Cry and started his business.
In the Vergil cut scene you see Vergil thrown in the blood lake where Dante fought Abyss in Mission 19. Suddenly the Prince of Darkness - Mundus from DMC1 appears. Vergil engages him in battle but the outcome isn't shown...
Sorry for the huge post, I hope I helped.