Both vehicles seem similar in terms of top speed, weight, armour and armament and were used to the end of the war. Was the Panzer IV chassis better than the Panzer III? Was the Panzer III chassis used for other vehicles like the StuG III earlier than the Panzer IV chassis was used for other vehicles?
Both vehicles seem similar in terms of top speed, weight, armour and armament
To an extent. Early on this is correct and true; they were practically built in parallel with one another; being 1935 and 1936 designs, respectively. By the end of the war the Panzer IV was a main battle tank, a medium meant to conduct the full breadth of missions; leading an attack, supporting an attack, exploitation, infantry support, etc. In regards to tank-on-tank combat , it superseded the Panzer III starting around 1943. Initially, the Panzer IV was meant to help the infantry while the Panzer IIIs engaged enemy armor. When the Germans began to run into M3s, M4s and T-34s the Panzer III began to show itself as less and less effective against these enemy medium tanks, and while its service life was extended with higher velocity cannons and additions/alterations to the armor casting, by mid 1943 it had been upgraded to its limit.
The Panzer IV began to take over the tank-killer role with the Ausf. F2; which put a high velocity 75 barrel in the turret. By 1944, the Panzer IV Ausf. H and the Panther were arming the battalions of a Panzer Regiment. This was opposed to the first battalion being Panzer IIIs and the second being Panzer IV prior to this.
The Panzer IV's chassis was capable of bearing greater ammo, weaponry and armor (the H to J nominally having 80mm on the glacis, in reality for the H 50 with 30 tacked on) becuase of its slightly bulkier chassis. This allowed its service life to be extended for a far longer time than the smaller and lighter Panzer III.
Was the Panzer III chassis used for other vehicles like the StuG III
Yes. Panzer III Chassis vehicles were in action alongside turreted Panzer IIIs. StuG IVs were far rarer than StuGIIIs, for example. The phasing out and decommissioning of Panzer IIIs also meant there were more of their chassis lying about for re-purposing. MarderIIIs, Befehlwagens (Command variants), the "N" (Howitzer equipped for close in support), and Armored Recovery vehicles, to name a few. To be sure Panzer IVs had many vehicles based on their chassis as well, but the actual numbers of such vehicles in service versus Panzer III chassis vehicles simply do not compete.
Most PanzerIV Chassis were used either for StuG IVs, or for the various Flakpanzer models there were seen in small amounts.