Heading east from Bali takes you across the Wallace Line and into the Indonesian province of West Nusa Tenggara, which includes the Gili Islands, Lombok and Sumbawa. Heading further east from Sumbawa takes you into East Nusa Tenggara, home to Flores, Sumba and West Timor - and others. While this is primarily an administrative border, the change from lush and over-popular Bali to Lombok and beyond is marked -- both socio-economically and environmentally.
The islands of Lombok and Sumbawa are both dominated by enormous volcanoes (Rinjani and Tambora respectively) and are arguably best known for their beaches and islands. Both, but particularly Sumbawa, are drier than Bali, with often savanna-covered islands sprinkled off their coastlines and without the extensive irrigated ricefields that Bali is so famous for. In dry season, which runs roughly from March/April to October, both can get really parched. They're best visited at the very tail end of wet season when you'll probably get some rain, but you'll see them at their lush and green best.
When compared to Bali, it would be fair to say that the tourism infrastructure in Lombok is adequate. In most destinations you'll find accommodation built with foreign tastes in mind (at least in theory!) and transport is relatively straightforward. On the Gili Islands of Gili Trawangan, Gili Meno and Gili Air, you could easily think you were still in Bali. In Sumbawa, despite a network of excellent roads, tourism infrastructure is still nascent, and very thin on the ground outside of the two surfing areas.
Another difference is religion. Lombok and Sumbawa are predominantly Muslim, and while Lombok has a significant Balinese Hindu minority, Sumbawa is almost universally Muslim. Food is simple, affordable and cheap -- don't expect the rich diversity of international cuisines you'll find in Bali! Alcohol is generally not sold openly outside of tourist areas, but hotels can often assist if you'll like to enjoy a beer at the end of the day.
Outside of Senggigi in the west and Kuta-Lombok in the south, Lombok sees a fraction of the foreign tourists Bali does. Sumbawa, outside of two surfing districts, sees almost none -- really, nobody. This is a shame, as Lombok and Sumbawa are rich in attractions and will be of considerable interest to repeat visitors to Indonesia. Beach lovers in particular are well catered for.
Lombok has one international airport (in the south at Praya) while Sumbawa has two domestic airports (Sumbawa Besar to the west and Bima to the east). Both are interconnected by car ferries and onwards to Flores in the east and Bali in the west. Both islands are in the same timezone as Bali, GMT +8.