Does Relativity make a difference on whether Earth revolves around the Sun or not? According to the theory of relativity there is no ultimate preferred frame of reference in our universe. It means that whenever there is motion between object A and object B, whether it is considered that object A is moving or object B is moving or both are moving depends upon your frame of reference.
If we follow this argument then what we study in books that the Earth is revolving around sun is actually not always correct but it should be correct only in a certain reference frame and it should also be correct to say that the sun is revolving around the earth according to frame of reference on earth. Why then do we say only earth revolves around the sun? Does it mean heliocentrism is not fully correct?
 A: Scientific theories can be thought of as models that take complex physical phenomena and seek to provide rules that explain the behaviour in simpler terms.
These models don't necessarily have to be perfect to be useful. For example, we know that Newton's law of gravity is not as precise as Einstein's General Relativity, but it is still good enough for most everyday purposes and the differences weren't even noticed for centuries.
The model that the Earth revolves around the Sun (and the Sun is fixed) is not very precise, but it is very simple. We can improve it by adding Newton's law of gravity and stating that the Earth and Sun revolve around a common centre of gravity. We can then add in the effect of the gravity of other planets, we can use relativity and even the effect of objects outside the solar system. But then the rules get a lot more complicated and perhaps we don't need that for what we are trying to understand.
So, yes the simple model of the Earth rotating around the Sun is not very accurate. But what we can say is that it is better than the simple model of the Sun rotating around the Earth. If we try to use this model then when we look at other planets we find they have very strange paths around the Earth (looping back on themselves) whereas if we consider the Sun as the centre then we find the planets orbit around the Sun.
So, yes, the idea of the Sun at the centre is a big simplification, but it is a useful one that can help our understanding of the solar system, whereas the idea of the Earth at the centre is not very useful for that.
A: All inertial motion is relative in relativity, not just any motion.
In STR, circular motion is always accelerated and this frame of reference is distinguished from inertial one.
In GR, the spacetime is curved and it happens that even revolving object might be inertial, which is more or less the case of objects in free fall, like Earths revolution around the Sun.
The GR is however local theory. This means, that frame of reference is usually physically meaningful only in small neighborhood of your position and the further away you are and the stronger and wilder the gravitational field, the more the frame of reference becomes just some coordinate system without any inherent meaning. This would mean that we indeed cannot speak about revolution of Earth around Sun without stating the reference frame.
But, in the case of Sun and Earth, the gravity is pretty weak and dominated by Sun. So in first approximation, you have just static field in vacuum of perfectly spherical object, which leads to Schwarzschild geometry. The thing is, that in this first approximation there is only the Sun in whole universe and it picks up special frame of reference - the one which is locked to it (and to distant stars which fixes rotation). So in the Solar system, the system itself has natural frame of reference and the statement about revolution of Earth is implicitly stated in this frame of reference. As it is natural, you do not need to talk about it explicitly.
If you go into more precision, you can keep the original frame of reference and just compute perturbation around it. So even in infinite precision theory, the frame as defined by first approximation is still the natural one and the statement about revolution keeps its meaning.
So in relativity - both STR and GR - the statement that Earth revolves around Sun is natural and we do not need to say in which frame of reference was this statement said.
A: 
it should also be correct to say that the sun is revolving around the earth according to frame of reference on earth. Why then do we say only earth revolves around the sun?

Technically if you are only considering an Earth-Sun system, both bodies will orbit about their center of mass. Since the sun is so massive, this essentially constitutes of the Earth orbiting around the sun, as seen from any inertial frame of reference.
Of course moving to a frame of reference where the Earth is stationary we will observe the sun to be moving around us. But in order to explain why we see this type of motion, and to explain why we are at rest, we need to bring in pseudo-forces that only "exist" in non-inertial frames.

Does it mean heliocentrism is not fully correct?

No, it does not mean this. In any inertial frame you will see all the planets moving in orbits about the sun. Even in the non-inertial frame where we see the sun moving around us we would not say that everything is revolving around us.

Ultimately, I think the issue here starts from the beginning. You are reading this

According to the theory of relativity there is no ultimate preferred frame of reference in our universe.

as

If two frames of reference disagree on something then one of them has to be wrong.

There is nothing wrong with saying that the Earth orbits around the sun, or that the sun is the center of the solar system. It's true in inertial frames, so it's a valid thing to say.
A: In SR, only inertial reference frames are equivalent, and so you can't tell which object moves only by experiments performed in those different reference frames. However this is not the case for non-inertial reference frames, which moves with acceleration. Earth rotates around sun, thus it has centripetal and centrifugal pseudo force acting on it, so this makes Earth a non-inertial reference frame cause now it has centripetal acceleration, which in principle can be measured. Common form of gravitational acceleration on Earth surface is :
$$ g= G\frac{M}{r^2}-\left(\frac{v_{e\perp}^2}{r}~\pm\frac{v_{s\perp}^2}{R}\right) $$
Where $M$ is Earth mass, $r$ is Earth radius, $R$ is Earth orbit radius, $v_{e\perp}$ - Earth rotation around own axis tangential speed and $v_{s\perp}$ is Earth rotation around Sun tangential speed. Sign $+$ or $-$ in formula depends on where are you are measuring gravitational acceleration  - in Earth side directed towards sun, or in the side directed outwards from the sun (then Earth axis rotational bulge effect adds to the Sun centrifugal force).
Centrifugal acceleration due to Earth rotation around own axis is small, about $0.03 ~m/s^2$. Centrifugal acceleration due to Earth orbiting Sun is even smaller, about $0.005 ~m/s^2$. But it's not zero and can be measured as variations in gravitation acceleration over time period of day (dependently on your relative radial position towards the Sun). If like you said not Earth would be rotating around Sun, but Sun instead around us - we would not get such third term in equation and would not experience mentioned gravity variations in time. Thus Earth/Sun reference frames are not identical to each other, and which is which- can be measured by local experiments in RF.
