First of all, a bit of clarification:
- The force (and therefore the acceleration) in a circular uniform motion is not constant, it changes in direction. However, it is true that, is the angular velocity $\omega=v/L$ is constant, the force's magnitude is constant. In the case of a pendulum, the force changes both magnitude and direction over time.
- The two motions (pendulum and circular) are both periodic but they are different in general.
Second of all: I wil call $\alpha$ and $\omega$ the angular acceleration and speed, $v$ the tangential speed and $L$ the length of the pendulum / radius of the circular motion.
Having set this, now the main difference between the two motions is that in a harmonic motion you have a constant change of energy from kinetic to potential, i.e. kinetic energy turns into potential energy (be it gravitational energy in a pendulum of elastic energy in a spring) and then in turn potential energy changes into kinetic energy. This is because there is an external (conservative) force which is doing some work on the system. In both the case of a pendulum and of circular motion, the tension of the string (or whatever centripetal force you are applying in circular motion) are not doing any work because they are always perpendicular to the displacement of the mass. Their only role is changing the direction of the velocity keeping it tangential at all times. In the case of a pendulum, however, you have gravity which is taking some of the kinetic energy and thus decelerating or accelerating the mass locally. Hence the change in speed therefore the acceleration therefore the change in force felt by the mass.
In the case of circular motion on the other hand, as there is in general no external force, the kinetic energy hence the speed hence the acceleration hence the force are (in magnitude) constant.
So for a circular motion you have, as equation,
$$\alpha = 0$$ $$\omega = v/L = const $$ so the equation for the angular acceleration reads
$$\alpha = d^2\theta/dt^2 = 0$$
So no external force / torque ($\alpha = I\tau = 0$ with $I$ moment of inertia of your mass - i.e. Newton's second law for rotations).
On the other hand, for a pendulum, you have an harmonic motion
$$\alpha = d^2\theta/dt^2 =-{g\over L}\theta$$
so you have an angle-dependent torque $\tau =-{g\over L}\theta$ that causes the pendulum to change speed. And it comes from gravity which "slows down / accelerates" the mass as it changes position differently (because the tension "steals" some of the gravity).
If you take a pendulum and put it on a frictionless horizontale plane (i.e. no gravity) and you give it a spin it will keep spinning at constant angular velocity as in circular motion.
So the motions are in general different. However, there are analogies i.e. circular motion and pendulum can be turned into one-another by adding/removing an external force.
If you take a circular motion and flip it so that now gravity is acting on it, then the speed (hence the acceleration and the force) will change over time as gravity acts on the mass as an external force, "stealing" some of the kinetic energy (which it later gives back). If your initial motion was "small" so that a small-angle approximation is valide, then your circular motion turns into a pendulum just by the addition of a restoring force (gravity). Otherwise, it is a more complex motion given by the equation for the angle
$$\alpha = d^2\theta/dt^2 = -{g\over L} sin(\theta)$$
(which becomes the pendulum equation if $\theta\approx0$ and $sin(\theta)\approx\theta$) which has a angle-dependent force - so the force will change over time - but you see already that if you have $g=0$ (i.e. no gravity) then
$$d^2\theta/dt^2 = 0$$ i.e. you recover a uniform motion given by $\omega=const$.
Here a small GIFs showing you the difference between the two motions: they are exactly the same, they start the same position and speed BUT one has gravity acting on it and the other one does not. So while they are both periodic (with same period) one oscillates (because of gravity pushing it back) and the other one does not. You can think of one being on plane, one being on plane with gravity.
