Kinematics is about the range of movement or change a system can undergo, or the state space in which it acts. Dynamics is about the movement it undergoes according to the laws of motion.
For example the kinematics of a rigid body in space describes its possible coordinate positions and orientations and the range of velocities and angular velocities etc. The dynamics describes how these would change under the influence of a given system of forces.
This means that conservation of energy and other quantities is dynamical because it only holds when the equations of motion are in effect.
Although kinematics and dynamics are most used in classical mechanics you can extend the idea to quantum mechanics where the kinematics are described by the phase space and operators, while the dynamics is the evolution under the influence of a given Hamiltonian.
It is traditional to regard the distinction between kinematics and dynamics as absolutely clear cut, but possibly the most important thing to understand about them is that this is not always the case. As a simple example consider the case of a particle that can move along a fixed track. You could regard the constraint that keeps it on the track as kinematical and only its actual motion along the track would be part of the dynamics, but we know that at a deeper level the particle is held on the track by dynamical forces.
Another example might be conservation of charge. If you consider the Dirac equation for a charged particle in the presence of an electromagnetic field, you find that charge is conserved only under the influence of the equations of motion. If you quantise the system the charge is given by the sum of the quantised charges on the positrons and electrons which can only be created and destroyed in pairs. It is possible to view this as a kinematic constraint with the dynmaics only accounting for the motion of the particles.
Perhaps the best example is in electrodynamics where a vector potential describes the field kinematics with the electric and magnetic fields being given by suitable derivatives. In this case the Maxwell equation that tell us that the magnetic field has zero divergence is kinematical because it follows without use of the equations of motion, but the divergence of the electric field is equal to the electric current according to the equations of motion. So some of Maxwell's equations are kinematic and some are dynamic. In a deeper theory these fields may be derived from a system which exhibits electromagnetic duality where magnetic monopoles act as sources for the magnetic field. In that case the kinematic and dynamic parts of the Maxwell equation are interchanged under the duality so we are forced to realize that the original distinction between kinematic and dynamic was an illusion.
In the final analysis the evolution of the universe does not make the same distinction between kinematic and dynamic that physicists do and it is important to appreciate that at a deeper level kinematics may turn out to be dynamics or vice-versa. So any attempt to define the difference is to some extent arbitrary and may not stand the test of time.