In a sense, yes. The fundamental design difference between an StG-44 and an FG-42 is the cartridge. Whereas the first uses the 7.92x33 mm Kurz (an intermediate cartridge), the latter uses 7.92x57 mm Mauser (a rifle round). This in turn leads to a number of differences in design and usage. Because it chambered a much more powerful round the FG-42 was significantly more difficult to produce as well as requiring either chrome-nickle-steel or manganese steel in the manufacturing process, both of which were in short supply. The FG-42 was also much more difficult to handle in fully automatic mode because of its heavy recoil and somewhat odd magazine placement. I believe a pressed a stamped steel version was made in small numbers late in the war but I am unsure about whether or not it suffered the same reliability issues early experiments with automatic rifles made from stamped steel firing rifles rounds suffered.

Now, the FG-42 was, to my knowledge, never considered for general distribution, it was always intended as a weapon for paratroopers and if memory serves few if any were ever issued to outside of the 2. Fallschirmjäger division and none outside the fallschirmjägers in general. As to its cost, the FG-42 was made in so many different iterations and in such small numbers that I don't believe the unit cost has ever been reliably estimated. If I had to hazard a guess I would say it would end up somewhere in the same range as the MG 42 and MG 34 so ~250-350 RM in 1944 (respectively) but this only on the overall similarties in design. (I want to stress that this is pure guesswork on my part. I have not found a single source on the overall production cost of the FG-42 but later designs do share many similarities with the MG 34/42 in its use of steel alloys receivers and stamped metal/wood parts)

The StG-44 on the other hand was a munch simpler design in many ways. Because it had a less powerful cartridge it could be made with pressed and stamped steal making it both cheaper, faster and easier to produce than the FG-42. It never had the power or long range precision of the FG-42 but the 7.92x33 Kurz cartridge is still more powerful than the MP 40's 9x19 mm pistol round or the russian counterpart for the PPSh's 7.62x25 mm Tokarev cartridges. In fact, for a relatively "simple" automatic weapon, the StG-44 proved remarkably accurate with an effective firing range of ~600 meters in semi-automatic mode (which is comparable to the FG-42) and about half that in fully automatic mode. It should however be noted that long range fully automatic firing was strongly discouraged, full auto was to be reserved for close engagements.

Basically, the StG-44 was cheaper, faster to produce, did not rely on materials that were in short supply, had a bigger magazine capacity and was almost as accurate. While adding another cartridge to the already strained supply lines of Nazi Germany was not ideal, it was still preferable over lacking a short/mid-range automatic weapon to counter the PPSh.

If you want more information on the FG 42 I highly recommend Thomas Dugelby's Death from Above - The German FG 42 Paratroop Rifle