On a private safari nobody else is in your vehicle. It is you, whoever you are travelling with, and your guide. That single difference changes the whole rhythm of the trip: you decide what time to leave camp, how long to sit with a leopard, when to stop for lunch, and whether today is a photography day or a slow day.
It suits families with young children, photographers who need to wait for the light, honeymooners, groups of friends travelling together, and anyone who would simply rather not negotiate the day with strangers. Because the vehicle is not shared, the cost of it rests on your party alone — which means a private safari costs more per person for two travellers than for six.
Everything else can be tuned to you as well. Departure dates are yours to choose, the route can be reshaped, and accommodation can be moved up or down in comfort without asking anyone else's permission.