Info from 2010: If you stick on the main road you will find it asphalted nearly all the way. Each meter from Cape Town to the Tanzanian/Kenyan border is asphalted (through Tanzania, Malawi, Mozambique). You will be amazed (roads have been financed and built by international charities). You will only do a few hours on dirt tracks in the whole trip.
So from the Tanzanian border to Nairobi you will find a dirt road, but you don’t need a 4x4 to go there if you drive slowly (might be wise to avoid the rainy season though, but even, no big deal - it’s a highway).
You will then find an asphalted road going through northern Kenya, but then the first (and only) tricky passage from Kenya to Ethiopia - a few kilometers. Westerners usually regroup to pass these kilometers in convoys, You just need to pass between tribal fights. These tribal guys might seem impressive (they have Kalashnikovs) but are not that dangerous (at this latitude you will find cultures that value honor, unlike the south where they tend to value more money).
You won’t have difficulties finding other westerners traveling along this long road: it is a highway for tourists. (All the way you will see rich guys with their amazing over-equipped 4x4s, which believe they are crossing the Martian Desert!)
I don’t know about asphalt in Ethiopia, but it doesn’t look that big a deal but … you might be stoned and insulted all the way by kids and teenagers, sometimes by adults (an Ethiopian custom).
As long as you are not Islamophobic and the arrogant-westerner type North Sudan will be your easiest part. North Sudanese were (by far) the friendliest culture I met on this road (from South Africa to France). Most of them are extremely poor but if you come from the south they will be the first African culture to see in you (the white guy) something other than a dollar with legs (a Mzungu). They don’t beg and don’t try to take advantage of your misfortunes ―which, coming from the south, seems incredible. (Some travelers find this culture difficult because they can’t buy these people's dignity, impose their culture or use their financial power to do whatever they want or impress the locals. The backpackers standard "disco, alcohol, cheap sex & drugs" is more difficult to find in this country than most other African countries.) These guys are also corrupt, but less so than in the south.
You may need to take the ferry with your car from Wadi Halfa to Aswan (no choice, it was forbidden to go by land).
Egypt was (and still is) a dictatorship and a military state. You may have to bribe policemen and the army to make it through the thousand checkpoints you will face. Keep in mind for your safety that (most) western policies support Egypt's current dictator (torture, execution) and some Egyptians don’t find it funny. But asphalt all the way up.
(Uganda is asphalted until its border with South Sudan, but South Sudan is a place you want to avoid: bad roads, aggressive people, weapons, tribal war, bandits, anarchy and oil.)
Globally your greatest difficulty won’t be the road state at all (you could do it with any cheap western city car) even passing through secondary roads, dirt tracks are not that bad, even during the rainy season (google search images of African cars/buses!). Your biggest difficulty won’t be to deal with those exotic things westerners talk about (lions and stuff) either, but the people (✲) and your skin color — if you are white (or "whiter"). (Don't forget that lions are the one in danger there). People will assume you are incredibly rich, or merely richer than they are (just because of your skin pigmentation) and will try all the way (below the Sahel) to take advantage of you (even those richer than you). It’s not a big deal if you are rich and ready to bribe all the way up (or down) but it might cost you a fortune (literally) on bakshish depending on your patience and commercial skills. And by doing so you make the life of the guys following after you difficult.
Using local transportation will definitely be much less troublesome and much cheaper (especially the visas!) even if you use a (relatively) luxurious way to travel. Local transportation seems (to me) also much safer and much, much less troublesome because you will waste much less time dealing with the African corrupted bureaucracy and you will at least avoid being extorted by military and "police" — among others — who will carefully watch you from every corner to "fine" or "tax" you for anything they might think of the moment you pass by.
(✲) if you plan to do something more interesting than just driving from one touristic attraction to another. If you step out of your car without trying to run away from locals or buying their smile you will find within social interaction the most impressive things you will ever see. But be prepared for a huge and tough ―but fascinating― psychological gap. If you cross Africa to hunt for clichés (such as wildlife) you might be disappointed — unless you pay for it. But if you go there to learn about mankind, you will be astonished.