What is the difference between computational model, model of computation and a computer?
A model of computation is an abstract device used to perform computation.

For example, Turing machines are a model of computation.
They allow operations such as reading a symbol on the tape, writing a symbol on the tape, move left/right...
Minsky machines are another model of computation.
They allow operations such as increment a counter, decrement a counter, test a counter for zero.
Models of computation are used to reason about algorithms, to answer questions like: is problem X realizable with the model of computation Y? If so, how many operations would be needed? 

Those models are abstract because they do not describe what the available operations are, not how they are realized.
A computer is a realization, an implementation of an abstract model of computation.
To ease their daily use, and to get efficiency from various technologies, computers have various sets of operations, and it is hard to describe an abstract model of computation for them.
Yet, all the operations they perform can be expressed in terms of e.g. Turing machines or Minsky machines operations.

Note that a Turing machine cannot exist for real: a Turing machine has an unbounded tape, so it would not fit in our Universe.
Similarly, the counters of a Minsky machine are supposed to be able to store arbitrarily high numbers.

From a strict theoretical point of view, since a computer has a finite fixed amount of memory, it is nothing more than a finite state automaton (with a large, yet fixed, number of states).
In practice, a computer is rather considered as a Turing machine, with a tape (memory) large enough not to run out.

To conclude, a computational model is a mathematical description of a complex system, often a physical one.
Computational models are mostly intended for simulation, to be run on real computers.