84fiero is SPOT ON.
Swaziland is a completely different culture and vibe. Given the choice between Swazi and Lesotho, Swaziland is the better choice. Lesotho struck me as "Little South Africa". Spend a couple hours reading up on the country and you'll be hooked.

Last month I drove from JNB to Durban, via Swaziland. (I had to pick up several dozen glass 'rubber ducks' from Ngwenya Glass Factory for wedding gifts. It also is a nice place for lunch and a cold beer. Just stay up-wind of the chocolate factory that is also on the grounds...or you'll be craving chocolate too!)

You will need your rental car agency to print you a letter stating you're taking their car out of SA. It's routine for them and they'll know what to do. Keep this letter (although I was never asked to show it on any of my cross-border drives in the last five years).

Getting across the border can be a bit confusing. It involves first stopping and getting an entry permit as you drive into the border area. You stay in your car for this, and the guard simply hands you a slip of paper on which he has written your license plate number. (You'll need to get this slip of paper stamped at the customs office.)
You'll then be directed to Customs, where you park and go inside; your passport is inspected and you have to pay a 50 Rand road permit. Get your receipt and your little slip stamped. Get back in you car and drive 50 meters to the border crossing. Go inside this building for your passport stamp, and the receipt from your 50 rand road permit is then collected. If so inclined, grab a free condom from the gift basket. Then, back into your car and drive to the border exit, where your little slip of paper with all the stamps is then collected.
On a good day, it takes 20 minutes. If it's a national holiday or long weekend, all bets are off. I've heard horror stories, but I've never seen one. The process is essentially the same for Lesotho. Leaving either country for South Africa is the same process, minus the 50-Rand fee.