OK, here's my experience. Main differences between the two variants are: the MS 250 C has the E2S recoil system, compression relief valve on the cylinder, different carb with a remote primer bulb, additional fuel lines and a fuel vent, plus a Quick Tensioner chain bar adjustment. The MS 250 has none of that.
I bought an MS-250C. After a while I got tired of the Quick Tensioner (OEM part 112300701008) bar loosening up and the disconnected feel of the E2S recoil start ( OEM part 1123 080 2108). So I bought an OEM recoil starter assembly for the MS 250 (1123 080 2104) and an aftermarket flywheel for the MS-250 (OEM Part 1123 400 1203) along with an aftermarket MS-250 bar tensioner assembly (OEM part 1123 007 1004) , bar studs (OEM part 1123 664 2400) , bar stud nuts (OEM part 0000 955 0804), and chain cover (OEM part 1123 640 1705) . I sourced all of my parts from Amazon and eBay, except for a woodruff key from Ace Hardware. Important addition, some of the aftermarket flywheels I saw on Amazon and eBay were out of phase - you have to view the picture to make sure the magnets on the flywheel are in the same position relative to the crankshaft key.
I was initially worried I would have to replace the coil and spark plug wire assembly, but that wasn't necessary. On the engine side you can go back and forth just by swapping the recoil starter assembly and flywheel. I should note that the OEM flywheels have a key cast into the shaft opening where some aftermarket parts do not, which was the case with mine. I bought a woodruff key from Ace Hardware and filed it down to fit. The compression relief valves, carbs, primer bulbs, vent lines, etc, or lack thereof. are not critical. Even though the coils between variants are different, it also doesn't matter. The chain bar adjustment systems are interchangeable also. So you can have an Easy 2 Start system or not, and a Quick Tensioner system or not, your choice. You can make an MS 250 C into a modified MS 250 and the reverse.
I did buy an Zama OEM carb rebuild kit for the carb, which was really all it took to get the saw running again. But I'm really pleased to have a near MS 250 that runs flat out.
Any questions I'm happy to answer. Cheers and happy woodcutting!