I have seen many papers that use admittance control for human robot collaboration but no one tells why admittance is the choice and not impedance. Is it something very trivial that I should know or what am I missing?
One aspect is that impedance control requires direct torque control of the actuators. Admittance control computes joint position or velocity commands in response to a measured force.
Many robot arms on the market simply don't allow external torque commands for the joints, only position and velocity commands. They undoubtedly have a torque control loop in the internal controller but don't expose any torque-control functionality to the user.
I would like to add a couple of things to the answer from @danzimmerman. It definitely depends on what your robot can measure and control. If you only can command position / velocities and measure forces, then the only option between both is to use admittance control.
Theoretically speaking, both can achieve the same performance. In practice however, both controllers have their own advantages depending on your specific task you want to achieve and especially the environment it will interact with.
Example: Imagine a manipulator holding a pen. Its task is to draw a straight line on something. This requires following a trajectory and maintaining a constant force against the wall.
Stiff Environment: Assume the wall is very stiff. At the moment of contact there will be an almost instantaneous feedback. An impedance controller would maintain the force by keeping the position error between the actual position and the internal models position. Since the wall is very stiff the impedance controller can quickly react to positional errors and provide a "good" performance. An admittance controller on the other hand would react to the change in forces. Since the change in forces will be very quick and could potentially overshoot due to the very stiff environment the admittance controller would probably result in a "bouncy" behavior and have a harder time to reach a "steady-state". After a high overshoot it would try to bounce back, and retry reaching the desired force etc...
Soft Environment: Imagine a very soft environment. At time of contact the force exerted at the end-effector would increase slowly. The admittance controller can slowly react to the change in forces and has the time to "adapt" to the displacement of the surface. Note that it will decelerate at the moment of first contact because there is a small change in force when pushing into the soft material. The impedance controller however, would not notice immediately that there is a change in force because if the material is very soft in the beginning it would maintain its speed perfectly fine. It would continue with the same speed until the material slowly stops its deformation and slow down the end-effector. Only then the impedance controller would get some feedback saying "hey, I notice a change in motion, this is the wall!".
Most of the times you will have an environment which is in between "soft" and "stiff". And your task might also require the manipulator the desired force in a specific amount of time. If you have the opportunity, definitely try both, observe the differences and choose according to your needs. As far as I can tell impedance controllers tend to be more "aggressive" in terms of force tracking, whereas admittance controllers are more "compliant". But this also depends on your tuning.
It is hard to formulate the specific behavior in words, but I hope I could give you some sort of intuition for it.