Based on the very successful airframe design and powerful engines of the MiG-29, as the designation implies the MiG-29M family is a very similar and a direct derivative of the original MiG-29A/C but essentially a whole new plane (in pretty much the same manner the Super Hornet/Rhino is different to a Legacy Hornet).
The most obvious shortcoming of the original MiG-29, namely its short range, was addressed in the new design by increasing fuel capacity with redesigned thicker LERXs and expansion of the dorsal ridge behind the cockpit in order to serve as a fuel tanks and space for avionics and countermeasures; the consequent increase in weight was countered with upgraded, now smokeless (a previous lever for criticism) engines, and more extensive use of composites (which also served as a radar-signature reducing measure, especially in the later MiG-35 variant).
As expected and in contrast to its predecessor the new Fulcrum is a genuine 4.5gen fighter which incorporates a FBW and fully digitized controls that further reduce weight and improve the already impressive high angle of attack and post-stall controlability of the original MiG-29 airframe. With the dramatic improvement of its avionics, radar and weapon software the M-family are not only significantly more capable BVR fighters but also genuine multi-role fighter/bombers.
During the 90s the only costumer for the MiG-29M was the Russian Air Force, with only a couple of tens sold to it. Its carrierborne version, the MiG-29K, achieved a "darling" status among the Russian Navy's carrier crews for its light weight, small size and overall practicality but at that time it lost in favor of the somewhat more powerful but way bigger, heavier and more demanding Su-33, the carrier version of the Su-27 family; a similar pattern was followed in both domestic and international markets throughout the late 90's and 00's, with the Su-27's equivalents of the MiG-29M, the Su-30 family, greatly overshadowing the MiG-29M/-33 family. Despite the lack of commercial success for the new Fulcrum and the seemingly bleak market conditions for them, Mikoyan kept investing and constantly upgrading the design up to this day ultimately culminating to the ambitious export-oriented MiG-35, an aircraft that retains all original Fulcrum's advantages with much improved technologies, including an AESA radar and even more powerful and longer living engine, that keep it extremely competitive, if not equal, to contemporary gen4.5++ and fifth gen fighters at a much more appealing price; similarly the MiG-29K made its triumphant comeback (now as a navalized version of the fully modernized MiG-35) taking its revenge upon the aging Su-33, which it now replaces, and scoring a significant export deal with the Hindu Navy. Now negotiations for export orders with the Russian Air Force/Navy, Serbia, Egypt, Syria and other less likely customers and its combination of high performance at a price that only the Gripen and the new F-16 variants can compete with, leave much ground for optimism in Mikoyan's marketing department.
sorry for talking that long in this one, the MiG-29 is among my favorites planes, if not my favorite, and I find the MiG-29M's story very emotionally charged I don't particularly like the MiG-39/MFI/1.44 but it will probably take longer, it's a long tale of struggle, unfulfilled dream and betrayal…
PS: "MiG-33" is the export designation of slightly improved MiG-29M variants throughout the 00's
PS2: Some of the improvements of the MiG-29M were retrofitted on older MiG-29C airframes as well as the new built MiG-29S/SMT "Fullback" that were essentially Legacy Fulcrums half-way upgraded to MiG-29M standard while retaining the older metallic airframe, conventional hydraulic controls and greater commonality of parts with the original Fulcrum-A/C in order to appeal to old MiG-29 buyers.