A budget safari does not give you a lesser Serengeti. The park fees, the guide, the vehicle and the wildlife are the same; what changes is where you sleep and how many people share the car.
Expect camping in dome tents or simple lodges just outside the park gates, a shared 4x4 with up to six travellers, and hearty camp-cooked meals rather than a menu. Vehicles are older but properly maintained and radio-equipped. Guides are the same qualified professionals who work at every level.
This is how most young travellers, backpackers and anyone travelling for a long stretch see Tanzania, and there is no shame whatsoever in it — plenty of people who could afford a lodge choose to camp because they prefer it.