Definition and Computational Model of EnumP Valiant [1] defined the $P$-enumerable problem in which he used the Counting Turing Machine Model: A counting Turing machine is a standard nondeterministic TM with an auxiliary output device that (magically) prints in binary notation on a special tape the number of accepting computations induced by the input. It has (worst-case) time complexity $f(n)$ if the longest accepting computation induced by the set of all inputs of size n takes $f(n)$ steps (when the TM is regarded as a standard nondeterministic machine with no auxiliary device).

A relation $R$ is $P$-enumerable iff there is a polynomial $p$ such that for all $x$ the set $\{y|R(x, y)\}$ can be enumerated in time $|\{y|R(x, y)\}|\cdot p(|x|)$.

However, in [2], they used the Random Access Machine (RAM) model to give the definition of $EnumP:$ Let $C$ be a set of binary predicates, $Enum · C$ is the set of problems $A$ such that $A \in C$.

It seems that $EnumP$ is more general than the definition of $P$-enumerable. Only the notations are easily confusing. 

Why do they use different models for the enumerating problem? What is the connection between the Counting Turing Machine and RAM model when deal with the enumeration problem? 

For the difference of TM and RAM can be referred to [3].

[1] Valiant, Leslie G. "The complexity of enumeration and reliability problems." SIAM Journal on Computing 8.3 (1979): 410-421.

[2] Capelli, Florent, and Yann Strozecki. "On The Complexity of Enumeration." arXiv preprint arXiv:1703.01928 (2017).

[3] https://cs.stackexchange.com/questions/22418/what-is-the-difference-between-ram-and-tm
First, you have to know that enumeration complexity is not as cleanly defined as other fields of complexity theory. People are still looking for the right definitions. Notions of reductions or completeness for example do not seem to generalize well. You will then have plenty of different names and notations for the same thing depending on the authors. For general overview of these notions, you can read the preliminaries of several thesis: Wojciech Kazana [4], Yann Strozecki [5] etc. The preprint [2] you cite is still a work in progress.

Now a clarification: Valiant does not use his Counting Turing Machines for defining P-enumerable. He uses them to define #P then reverts back to deterministic TM and defines P-enumerable. He is not really interested in enumeration complexity. He only vaguely defines it (he says that the set can be enumerated without defining it formally) to show that some #P-complete problems can still be enumerated (using fact 7, also known as "flashlight method" for enumeration).

Finally the answer. EnumP and P-enumerable are different indeed but not comparable. EnumP only says that the predicate you want to enumerate behave "nicely", that is, you can verify in PTIME that the outputs of your algorithms are indeed solutions. It does not says anything on the complexity of the problem of enumerating.

The notion of P-enumerable of Valiant corresponds to what is called Output polynomial algorithm, that is, you can enumerate the set in time polynomial in its size and the input size. This is what is called $OutputP^F$ in [2], as the "uniformity" condition (that the predicate should be in EnumP) is not present in Valiant's work. For example $SAT(\phi) = \{\tau \mid \tau \models \phi \}$ is in EnumP but not P-enumerable. See Proposition 7 of [2] for example to see a problem P-enumerable but not in EnumP.

As for the choice of RAM over TM for enumeration, the polynomial time is usually not affected by this choice, so this does not make a huge difference for complexity classes. However, such a choice is common in enumeration complexity as people are often interested in constant or linear delay, which is more robust on RAM than on TM (see for example, Wojciech Kazana thesis [4] for constant delay or the work of Étienne Grandjean for consideration on the class of linear time solvable problem). 

[4] http://www.lsv.ens-cachan.fr/~kazana/PAPERS/Wojciech_Kazana_16Septembre2013_QueryEvaluationWithConstantDelay.pdf

[5] http://www.prism.uvsq.fr/~ystr/these_strozecki