Why is there a need for the concept of energy if we have the concept of momentum?

Here is a thought experiment that should convince you that it's necessary to consider energy and not just momentum. Suppose you stand on a train track in between two identical trains traveling at the same speed. One of them approaches you from your left, and the other from your right. The trains are moving in such a way that they will both collide with you at the same instant.

Since the trains have the same mass and speed but are traveling in opposite directions, they carry zero total momentum. When they hit you, they will not tend to accelerate you to the left or to the right — you will stay right in place. In spite of their lack of net momentum, the trains do carry kinetic energy. When they collide with you, they will transfer some of this kinetic energy to your body. Anyone watching will clearly see the effect of this energy, although you yourself will probably not be able to observe it.

Already some good answers here, but also let's add the following important idea which no one has mentioned yet.

Suppose two particles are in a collision. The masses are $m_1$, $m_2$, the initial velocities are ${\bf u}_1$ and ${\bf u}_2$, the final velocities are ${\bf v}_1$ and ${\bf v}_2$. Then conservation of momentum tells us $$ m_1 {\bf u}_1 + m_2 {\bf u}_2 = m_1 {\bf v}_1 + m_2 {\bf v}_2. $$ That is a useful and important result, but it does not completely tell us what will happen. If the masses and the initial velocities are known, for example, then there would be infinitely many different combinations of ${\bf v}_1$ and ${\bf v}_2$ which could satisfy this equation.

Now let's bring in conservation of energy, assuming no energy is converted into other forms such as heat. Then we have $$ \frac{1}{2}m_1 u^2_1 + \frac{1}{2}m_2 u^2_2 = \frac{1}{2}m_1 v^2_1 + \frac{1}{2} m_2 v^2_2. $$ Now we have some new information which was not included in the momentum equation. In fact, in a one dimensional case these two equations are sufficient to pin down the final velocities completely, and in the three-dimensional case almost completely (up to rotations in the CM frame; see below). This shows that energy and momentum are furnishing different insights, both of which help to understand what is going on. Neither can replace the other.

There are plenty of other things one might also say. The most important is the connection between energy and time on the one hand, and between momentum and position on the other, but other answers have already mentioned that. It may also interest you to know that the two most important equations in quantum theory are a relationship between energy and development in time (Schrodinger's equation) and a relationship between momentum and position (the position, momentum commutator).

Further info

The general two-body collision can be analysed in the CM frame (variously called centre of mass frame; centre of momentum frame; zero momentum frame). This is the frame where the total momentum (both before and after the collision) is zero. The conservation laws fix the sizes but not the directions of the final velocities in this frame, except to say that the directions are opposite to one another.

To me, momentum is more fundamental than energy ... In short, I want to know the physical difference b/w momentum and energy.

In modern physics the most fundamental facts have to do with symmetries. From Noether’s theorem any differentiable symmetry in the laws of physics corresponds to a conserved quantity.

Momentum is the conserved quantity associated with spatial translation symmetry. In other words, the laws of physics are the same here and there, therefore there is a corresponding conserved quantity which we call momentum.

Energy is the conserved quantity associated with time translation symmetry. In other words, the laws of physics are the same yesterday and today, therefore there is a corresponding conserved quantity which we call energy.

In modern physics neither time nor space is prioritized over the other, but together they are unified in a single overall framework called spacetime. In that framework energy and momentum are the timelike and spacelike parts of a single overall conserved quantity called the four-momentum which is a four-dimensional vector $(E/c,p_x,p_y,p_z)$. It is incorrect to assert the preeminence of either over the other. They are non-redundant conserved quantities.