what is the real difference between traveling salesman problem (TSP) and vehicle routing problem (VRP)?
The Vehicle Routing Problem was introduced in G. B. Dantzig and J. H. Ramser, The Truck Dispatching Problem, Management Science Vol. 6, No. 1 (Oct., 1959), pp. 80-91.

The authors underline the differences with TSP in this way:


... The "truck dispatching problem" formulated in this paper may be considered as a generalization of the TSP ...

... The salesman may be required to return to the "terminal point" whenever he has contated $m$ of the $n-1$ remaining points, $m$ being a divisor of $n-1$. For given $n$ and $m$ the problem is to find loops such that all loops have a specified point in common and total loop length is a minimum. Since the loops have one point in commom, this problem may be called the "Clover Leaf Problem"...

... The TSP may also be generalized by imposing the condition that specified deliveries $q_i$ be made at every point $P_i$ (excepting the terminal point). If the capacity of the carrier $C$ is greater than $\sum_i q_i$, the probelm is formally identical with the TSP in its original form since the carrier can serve every delivery point on one trip which links all the points...


In the simplest VRP formulation, all trucks (vehicles) have the same capacity and only one product is to be delivered to each point $P_i$. Other common constraints are: time constraints (or total length of each route), time windows, precedence relations between points.

To summarize: the main difference between a TSP and VRP is that the salesman must return to the starting location after some points have been visited. 

For what regards "It seems relatively easy to model TSR in the form of VRP and likewise inversely."; the reduction from TSP to VRP is immediate, the opposite direction VRP $\leq_m^p$ TSP is surely more complex (and probably it requires other intermediate reductions).