What is the difference between mean free path and intermolecular distance? Why is the mean free path not be equal to the intermolecular distance?
A particle moving in a particular direction should strike the object in that direction after the traveling the same distance as the distance between them initially.
 A: The difference lies in the cross section of the particles.
Consider two equally large volumes containing an equal amount of particles, but the particles in volume A are twice the radius of the particles in volume B. In this case, the inter-particle distance is the same in both volumes, but the mean free path in volume B is four times the mean free path in volume A.

If the density is $n$, and the cross section of the particles is $\sigma$, then
$$\mathbf{mean \,interparticle\, distance\!\!:}\,\qquad \langle r \rangle \sim \frac{1}{n^{1/3}} \qquad\mathrm{(independent \,of \,cross\,section)},
$$
while
$$
\mathbf{mean \,free\, path\!\!:}\,\qquad \ell = \frac{1}{\sigma n} \qquad\mathrm{(inversely \,proportional \,to \,cross \,section)}.
$$
For a real world example, consider a Lyman $\alpha$ photon that first enters an HII cloud of ionized hydrogen atoms with a density of $1\,\mathrm{cm}^{-3}$, and subsequently an HI cloud of neutral hydrogen atoms with the same density. The clouds have the same inter-particle distance, but the while the HII cloud is transparent to the Lyman $\alpha$ photon (because the probability of interaction is virtually zero), the HI cloud will likely scatter the Lyman $\alpha$ photon multiple times.
A: There's a few things to point out. @lemon already pointed out one of them in the comment -- it is possible for a molecule to move and end up going between other molecules and missing them. So even though they start out, say, equally spaced (think like an equilateral triangle), it's possible for a molecule to move between the other two rather than directly at it.
The other is to remember that all the molecules are moving. Consider a very simplified case where you have two molecules on the X axis, initially a distance Y apart. They are both moving at +Z on the axis. Their mean free path is infinite -- they will never collide -- but their intermolecular distance is constant. They are just following each other. 
So, all of this is to say that the molecules are all moving at the same time and in random directions. It is not as if you have a single molecule moving and the rest are frozen. Nor are they guaranteed to be moving directly towards one another. The mean free path is how long a molecule travels before it hits something on average, while the intermolecular distance is the mean spacing between the molecules without consideration for their direction of motion.
