What distinguished physical and pseudo-forces?

Because it is different in different reference frames. You would expect a universal force to be the same regardless of how you look at it.

Also, it is not easy to figure out the source of the force.* You feel like being pushed by something you can't identify, which appears to break with Newton's 3rd law. That is an indicator (although not certain proof) of it possibly being an effect we feel due to something else than forces.

Finally, you won't feel the centrifugal force if you close your eyes.** This indicates that it might be an illusion, a mind trick, due to the surroundings changing.

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*^{Think centrifugal forces, Euler forces and the push you feel when standing in a braking bus, and then try to look for the source.}

** ^{Sure, in the turning car or braking bus you feel the seat underneath you pulling sideways or backwards. You feel the car door or bus rail slamming into you, pushing you. But try jumping just before the turn or the braking - you won't feel anything and with your eyes closed you won't notice any changes at all. Not until you touch something again, that is.}

Pseudo forces aren't physical forces because we, humans introduced them in order for Newton's 2nd law to work properly in non-inertial frame of references. Basically it's mathematical correctional term,very similar to cosmological constant introduced by Einstein in general relativity equations (if you know/heard of about that constant). For more details, you can check my other answer at : If an object is free falling under gravity, then where is the pseudo force applied?

Real forces like gravity, are 'real' because we see and are influenced by the effects like by gravity. So much so, that our evolution is based on to counter gravity of our earth and thrive on land.

Note : cosmological constant and pseudo forces are not related or similar things. They are merely mathematical correctional terms in their respective theories in order for theory to work properly.

When you spin the ball attached to a string as you stated, what we call centrifugal force arises from the ball's inertial tendency to move away in a straight line per Newton's first law of motion. The string provides the centripetal force that keeps the ball from doing so. So we see the centrifugal and centripetal forces as resulting from the ball's inertial resistance to a change in acceleration rather than as direct forces alone.

Real forces satisfy two fundamental requisites

1. They do not depend on the reference frame

2. They satisfy the third law of classical dynamics

Pseudoforces violate both requirements. Indeed, they appear only in some reference frames called, non-inertial in addition to real forces and they are added just to impose the validity of $vec{F}=mvec{a}$ also in those reference frames. Secondly, all real forces always arise in pairs in the Newtonian formulation of mechanics: the body B exerts a force $vec{F}$ on the body A and simultaneously the body A exerts the force $-vec{F}$ on the body B. If $vec{F}$ is a pseudoforce acting on $A$, there is no $B$ and there is no $-vec{F}$ acting on it.

(I stress that the notion of pseudoforce is proper of Newton's formulation of classical mechanics. When passing to more general relativistic formulations, that distinction between forces and pseudoforces is not so sharp also because the language is different, and pseudoforces share the same geometric nature of the gravitational interaction: they are no longer "forces". So that it is difficult to directly compare the two scenarios.)

This is a comment really.

Thinking of Newoton's force, F=ma as really F=dp/dt where p is the momentum vector gives the perspective where what is called a fictitious force and a force are the same thing. It also makes momentum important and makes force liable to the conservation of momentum law. This perspective leads naturally to the use of the term "force" in particle physics, interpreting Feynman diagrams, where there is a dp/dt at the vertices. It is the first order exchanges of gauge bosons that characterizes the "forces" of the standard model into electromagnetic , strong and weak.

A pseudoforce indicates a force that would be absent in a different reference frame. A simple example is a linearly accelerated frame. If you weigh yourself in an accelerating elevator, your weight will deviate from its value in Earth frame. The spring inside the scale obeys Hooke's law, F=kx, tells you that x and thus the force exerted on it is different upon acceleration. This is a real force change, not a mathematical device. Why do we call this a pseudo force? Because we know that there is a simpler reference frame in which it is absent. By this reasoning gravity itself can be considered a pseudoforce. The simpler frame then is free fall.

(This is a simplistic answer,that may be loose in language, and not universal. But may help a lot)

Newtonian mechanics is primarily formulated (in a simplostic sense) for inertial frames.

In a non-inertial (non relativistic) reference frame, the frame itself inherently incorporates its own acceleration/rotation. But you don't see it, really. When you walk on the rotating earth, or corner in a car, you feel some things but you don't necessarily directly perceive with your senses the non-inertial nature of your own reference frame.

So the effects you do perceive - that the car door pushes you inward, or the cars motion seems to throw you outward, or the non Newtonian precision measurement on a rotating Earth - in a way,the pseudo forces are manifestations, or (misleading!) perceptions, that either account for or offset the less easily perceived non Newtonian aspect of the frame.

Why are some forces considered pseudo-forces while some are considered real or physical forces?

There are a variety of theoretical distinctions as listed in the other answers. However, since physics is an experimental science I prefer the practical/operational distinction:

Real forces can be detected with accelerometers and pseudo forces cannot.

However, a person certainly "feels" the pseudo-force such as in a car that is rounding a corner. How can it be proved, then, that the centrifugal force is not a real force?

Actually, this is incorrect. Removing all of the complicated neural processing and psychology, just stick with accelerometers as indicators of what is actually “felt”.

When you are in a car rounding a corner your accelerometer reads an inward acceleration. Thus the only force that is actually felt is the centripetal force. The centrifugal force is not detected. If it were then the accelerometer would read 0. So both the direction and the magnitude of the accelerometer reading contradict the claim that the centrifugal force is a real force that is “felt”.

In Newtonian physics, force is equal to the rate of change of momentum. In Lagrangian mechanics, we assert a function of position and velocity that governs the physics called the 'Lagrangian', and then 'force' is the partial derivative of the Langrangian with respect to position, and 'momentum' the partial derivative with respect to velocity. It looks a bit like:

$frac{partial L}{partial x}=frac{d}{dt}frac{partial L}{partial dot x}$ where $frac{partial L}{partial x}=F$ and $frac{partial L}{partial dot x}=mv$

But Lagrangian mechanics allows positions $x$ to be expressed in non-inertial and curved coordinate systems (called 'generalised coordinates'), in which the relationship of the partial derivative to velocities in the flat, inertial coordinate system involves some extra terms. In an inertial reference frame (and assuming constant mass), these extra terms disappear and you just differentiate velocity to get acceleration (i.e. $F = ma$). In non-inertial or curved coordinates, you get extra terms on the right hand side, because you're differentiating with respect to something that itself varies. The trick with 'inertial forces' is just to transfer these extra terms over to the left hand side and call them part of the force. We're used to the form of the force being arbitrary, more so than the formula for acceleration.

Newtonian physics asserts as a postulate the existence of a set of inertial reference frames where these curvy extra terms vanish. It's not the only viewpoint on the matter, though.

In General Relativity, we allow much more freedom with regard to coordinate systems. If we consider a rock spinning in a static universe full of distant stars, Mach's principle suggests that this should be equivalent to a static rock in a universe with the distant stars spinning around it. And indeed, General Relativity predicts that there should be a gravitomagnetic force (a component of gravity analogous to the magnetic force in electromagnetism) that arises when there is a flow or current of mass, and in the case of the spinning universe turns out to match the centrifugal force. (See here.) Thus, the supposedly fictitious centrifugal force is just the gravitational force exerted by a spinning universe, and not fictitious at all! (Or at least, no more fictitious than gravity.)

In a sense, General Relativity pulls the same trick in reverse - it transfers all the forces over to the right-hand side of the equation and calls them curvature, leaving no 'real' forces. All particles follow free-fall paths through curved spacetime, and all forces are just aspects of curvature.

Of course, that doesn't apply in Newtonian mechanics, that asserts a simple inverse-square law for gravity, and a God-given set of inertial frames. Forces that can't be made to vanish and remain in the inertial frames are classified as 'real'.