Ok momentum is not KE, but one can calculate the amount of KE stopping the object and measureing how high it raised a weight, or now much frictional heat it generated. If momentum would be a different number, then how do you measure it? What is momentum? KE will also travel in the same direction until a force is acted upon it, so what is the purpose of having a term called momentum?
Momentum is mass times velocity, typically written p=mv. Kinetic Energy is mv^2/2. In any interaction between two masses momentum is always conserved. Energy is conserved also, but it can change forms from kinetic energy to heat energy for example. Or the kinetic energy can be stored in a spring or as potential energy. So even though energy is always conserved, it does not have to be conserved as kinetic energy. But regardless of the interaction momentum is conserved.
Another difference between momentum and KE is that momentum is a vector (has magnitude and direction) whereas KE is a scalar (magnitude only).
If momentum and KE is conserved, but KE can be conserved as PE like in a spring, then what happened to the momentum of the object after it hit the spring? It's not conserved like the KE is, and there's no term for potental momentum. KE and momentum must match up meaning where's there's some KE there must be some momentum, and visversa. But momentum is not conserved when a spring is compressed.
When talking about conservation of momentum in this case, you have to include the momentum of whatever the spring is attached to. If the spring is effectively attached to the Earth, then when the object collides with the spring and compresses it, the Earth recoils with a very very very very small velocity because of its very very very very large mass.
Ok then take two objects of equal mass in space. They fly at each other and hit a spring inbetween them. Energy is conserved because KE turns into PE. But for the time the objects are slowing down, stopped, and reaccelerating, their momentum is not conserved.
Momentum is conserved, because as it has been indicated, momentum is a vector quantity. To put a coordinate system on your example, two objects of equal mass flying in space. If one is flying in the +x direction, and the other in the -x direction, the total momentum of the system is exactly zero. So, when they hit the spring, we expect momentum to remain the same (zero), and it is.
So if a light object pushes off a heavy object, then they're moving in opposite directions and with the same KE, but do they have different momentums?
Momentum is always conserved for any isolated system. This is a direct consequence of Newton's 3rd law.
The objects do have different momenta, because they will be moving in opposite directions. The magnitudes of the momenta will be the same though. I'm not sure if the kinetic energies necessarily have to be the same, though. (This would mean that the distance the force of one objcet on the other is applied through is the same for both objects, and this doesn't necessarily seem intuitive to me.)