The biggest difference between the Machame and Lemosho Routes is their length: Lemosho is about 8km (5miles) longer. Therefore, the Lemosho Route takes one more day to complete: While Machame can be completed in 6 or 7 days, Lemosho takes 7 or 8 days.
Another difference is the starting location: Machame starts in the south, not far from Moshi, while Lemosho starts in the west. This makes the Lemosho Route a longer drive on your first day if you arrive from Moshi, therefore also slightly more expensive.
As both routes join mid-way, they are otherwise identical and offer the same scenic highlights further up the mountain.