Why $(apparently)$ is the rotation of Earth not congruent/similar to Uniform Circular Motion? Okay, so this may seem a very "simple" question.
I quote few lines from one website:

"We can’t feel Earth’s rotation or spin because we’re all moving with it, at the same constant speed."
"Why don’t we feel Earth rotating, or spinning, on its axis? It’s because Earth spins steadily – and moves at a constant rate in orbit around the sun – carrying you as a passenger right along with it."

Okay, I get the fact that the ground, the atmosphere and all the "things" on Earth are moving at the same rate. It is almost like we are travelling in a train or an airplane, where we can walk normally as if on the ground, without feeling anything unusual.
But, my "Physics" intuition seems to contradict this explanation. This is how I view this problem :
We dwell on the ground and move with the Earth, very much like in a "$2$-D Circular Motion", with the radius of the Earth as the radius of the "circle". But the velocity changes every instant as its direction (which is tangent to the circle at the given point) changes constantly. The velocity is changing --- So there must be some acceleration , as Uniform Circular Motion is always an accelerated motion.
So, I see that there is "some acceleration". According to Newton's Equation $F_{\text{net}}=m.a$, we must be experiencing some Force, which is required to provide the necessary Centripetal Acceleration$(=\dfrac{v^2}{r})$.
So, in short, I state my Problem as:

When we ride a Merry-Go-Round, we feel a "certain" force pushing on us, acting radially away from the centre. Why don't we feel the same "certain" force in case of Rotation of Earth?

EDIT : Can anyone please remove that "Duplicate" mark ? My question is not specifically about "Why don't we feel the Earth's rotation ?".
It is rather about "Why (apparently) is Earth's Rotation not congruent/similar to some normal Uniform Circular Motion (as in a Merry-Go-Round) ?". Surely, I will re-word the question to be more specific.
 A: Your reasoning is absolutely correct. We do feel a centrifugal force owing to the fact that we are in a frame that is performing the rotational motion. This is the reason why the weight of an object will be slightly more near the poles as compared to near the equator. Because the centrifugal forces will be maximum near the equator but will be vanishing near the pole. But this effect is quite small as compared to the gravitational pull of the earth. This is why we don't feel it in any obvious or trivial way in day-to-day life. 
The centrifugal accelerations produced in the merry-go-round (or giant wheels) are quite large as compared to the centrifugal accelerations produced due to the rotation of the earth. Thus, they appear quite noticeable. Also, the net direction of the centrifugal acceleration keeps changing its direction because the direction of the centrifugal acceleration due to the rotation of the merry-go-round keeps changing its direction (in the frame of the earth). In the case of the motion of the earth, the centrifugal force doesn't change its direction (in the frame of the earth).
A: You do "feel" the rotation it that the reading on a spring balance does vary between the Equator and one of the poles being larger at the poles than at the equator.
You do not notice the effect because it is so small - about $0.02 \,\rm Nkg^{-1}$ in $9.81 \,\rm Nkg^{-1}$.  
Another example is the circulation of large air currents in the atmosphere the direction of which are determined by the rotation of the Earth.
These are due to the Coriolis force.

A: Actually, you are experiencing a centrifugal force right now. How would you know if you were feeling it? Forces are vectors. You think you're only experiencing gravity right now, but in reality you're experiencing a net force which is the sum of gravity and centrifugal force. It is because of this centrifugal force that the direction of vertical is slightly altered. People at the equator feel less weight because the centrifugal force there is maximum and is in the complete opposite direction of gravity.
Please note that it is always the net force that you experience.
Actually, gravity is far greater than the centrifugal force we experience. At equator, centrifugal force opposite to gravity, so the only contribution of the centrifugal force is decreasing the net downward force that we feel by a small amount. At other places (except poles where there is no centrifugal force), the centrifugal force acts at an angle to gravity, so in addition to decreasing effect of gravity, it alters the vertical too.
A: You're right, and this is a great question. The centripetal force is provided by gravity on Earth. If a person is $80kg$, however, it is a very small force, $\frac{80*465^2}{6.37*10^6} = 2.71N$, which is actually a discrepancy on your bathroom scale due to the Earth's rotation!
However, if you're wondering why the rotation doesn't make us feel dizzy, imagine you're spinning in place. What's your linear velocity? $0\ m/s$, as you're still. But you're still feeling dizzy. This is because of a high angular velocity $\omega$, and that's what we really feel. But $\omega$ on earth is very small -  $7.2921159 × 10−5\ rad/s$, so we feel pretty stationary.
A: Because centrifugal force does act upon you.
When you're standing on a plane surface (in your reference not earth , earth is oblate). This centrifugal force is known as normal reaction. 
For you to be in constant state and not fly away or sink in ground, the normal reaction balances your weight. 
Comparison with merry go round is not apt one as on merry go round your weight is too big while the force that keeps you with the base plate  is very small and variable depending on how you sit/stand (frictional force).
In case you earth you are the size of a speck. Huge forces act upon you to balance out each other.
While you feel centrifugal force on merry go round the small speck of dust on it usually doesn't (unless you make the base plate too smooth, less friction)
The dust particle always stays on the ride of fun :)
