This question came to my head as I read the following quote by an unnamed WW2 veteran from "Second-hand time" by Svetlana Alexievich:
They sent reinforcements, but after two or three days, they were dead, almost all of them. A few survived. They were lucky! It was only until late 1943 when we had learned how to wage war. That's when we started to fight the right way. We no longer had as many casualties...
While I've heard this fact many times, I've never really reflected how this would be felt or experienced by the individual soldier.
It would be a very different army, but it would also depend on which branch you were serving in. Fighting in a Tank Army was very different from fighting in a Rifle Division.
As a general rule however these general rules would apply:
Command & Control was more consistent. A big issue in 1941 was that most Soviet units simply did not have effective command & control due to the strategic surprise achieved by the Germans. By 1943 it was much less of an issue.
A more balanced mindset between offense and defense. David Glantz took a look at the Soviet manual in 1941 and found only a handful of pages devoted to defensive operations. By 1943 defense was no longer being ignored and was half the manual. Whatever arm you were in, there would be more emphasis on how to fight defensively instead of just attacking - hence the impressive lines of trenches and mines at battles such as Kursk which frustrated the Germans.
Less centralization of key support assets - particularly tanks, artillery, and engineers. The losses in 1941 had caused many Soviet units to be stripped of all-important support units, which were instead concentrated in army-level HQ. By 1943 however the Soviets had regained confidence and were beginning to attach support units to frontline units again. This is why, for instance, some Rifle Divisions in Kursk had attached tank brigades supporting them.
More readily available supplies and ammunition, in part due to the arrival of Lend-Lease supply trucks. The Red Army despite its size had in fact shot fewer rounds of ammunition and used less fuel in 1941 and 1942 than the German Army. The trend began to reverse in 1943 due to both increased production and improved logistics.
Equipment should have also changed markedly for tank units, albeit not as often imagined. A 1941 tank Division would have been mostly equipped with light tanks such as BT-7s or T-26s. By 1943 they would have been revamped with either T-70 tanks or T-34 mediums. The SU-76 assault gun would also be much more common at this point. Infantry equipment would not have changed as dramatically however.
Edit: Additions.