Whereas both are maximum operating limit speeds, one is based on indicated airspeed and the other on mach. For instance, in one aircraft on a standard day, the Vmo may be the limit until reaching 13,000 MSL when the Mmo would be less than Vmo and would become limiting from there on up. While you might have a Vmo of 275, the Mmo might limit you to an indicated airspeed of 190 or so indicated at 30,000.
Both MMO & VMO are "redline" airspeeds. The difference is how each is defined, and when each one becomes a factor.
VMO is defined in terms of indicated (or calibrated) airspeed -- so many knots. It can be based on how much stress the speed puts on the airframe itself, or how much "G" can be put on the airframe at that speed (i.e. it becomes too easy to overstress the plane in turbulence or with slightly overaggressive control applications), or other things. It is a player at low altitudes, and in light airplanes, it is the only limit you worry about (which then makes it VNE, iirc).
MMO is defined in terms of a limiting mach number, and is based on what happens to airflow around the aircraft at that speed. If the jet is flying at, say, .82 mach, some of the airflow around the plane is moving faster, and will start to produce shock waves or buffet as it approaches transonic velocities.
Since Mach # is a function of TRUE, not INDICATED, airspeed, you rarely start to approach MMO at low altitude -- your IAS & TAS are about the same, and you will reach VMO before you get an especially high mach #. As you go up, however, for the same IAS, your TAS goes up, as does your mach #. Sooner or later, even keeping your indicated airspeed in the "middle of the green arc" (so to speak), your mach # will increase to the point that you reach MMO.
On the 737, the airspeed indicator has a barber pole that moves so as to always indicate the relevant redline speed. Down low, it's up at 340 KIAS (i.e. VMO). As you climb, it moves to lower & lower indicated speeds, reflecting the IAS that will yield MMO at that moment. As you descend again, it goes back up. At FL410, the barber pole may be around 260, and the typical profile is to fly .78M at altitude & in the descent, until that .78M is equal to 280 KIAS, and from that point on maintain 280 KIAS for the remainder of the descent. That way, you keep a respectful distance from both MMO and VMO (along with various other benefits from that profile -- ride quality, fuel economy, etc).
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