Odahviing takes a moment to appear when you shout for him, but he is physically stronger and will breathe fire fairly often. He will also stay with you until there are no enemies nearby.
Durnehviir does not do much damage, but can and will use the Drain Vitality and Soul Tear shouts, as well as summoning undead to fight with you. He will stay with you even if there are no enemies around, but he has a time limit. He appears instantly when shouted for though, since it's basically a conjuration shout.
Their is no difference, they both svck an equal amount of svcketry and non straightness.
Then you have not used them. During the Forsworn Conspiacy quest, I was being chased by roughly 25 guards, so I ran out of Markarth into the open field in front of the stables and called Odahviing. He roared, landed in the middle of them, and went crazy. Six of them broke away and came after me, and by the time I had dealt with those, he was casually tossing the last of his enemies to the side.
Both of them are very powerful, and are most useful when you are being attacked by a large group in an open area.