i think the two of them would retain their core personalities. ie dante still the fun loving guy and vergil the more serious one. but their intensities would be different.
ie dante would be insane and find everything funny and dynamic sortta like the joker. as being unable to cope with the loss of sparda and eva his mind descended into madness as an escape. he wants power to destroy because thats how he sees the world as. a constant series of destruction. so in someway hes saving people by ending the cycle of chaos once and for all.his fighting style would be showy, erratic and completely unpredictable.
Vergil would be more laid back. think dante in his more serious moments. he accepts that somethings are simply inevitable and wants to keep the world safe so no one needs to go what he went through. he cracks the odd joke here and then but over all keeps his attitude serious. hes a pacifist at heart, if he could he'd talk the deomons down instead of fighting them but he never shys from a battle. when he meets dante he begs him to stop and let him get help. to no avail. Vergils fighting style is linear and swift. he wastes no movements so each attack is a perfectly placed strike with devastating affect