Having just started to learn Forcing and having looked though the large number of Forcing questions on StackExchange, I apologize if I overlooked the following 'overview' novice question:
Enderton 2nd Edition 2001 p135 gives a way of constructing a countable model M for a First Order countable assumed satisfiable/consistent theory (as part of the Completeness Proof). It looks e.g. to add just a single constant for each sentence S of a first order language to witness each S of the form $\exists$xU(x) where U(x) is an expression in the language. The method 'looks' to be possible to be done by a computer program.
Presumably Forcing such as that in Kunen 10th Impression 2006 Chapter VII could be applied to the model M, if the language is taken to be ZFC, to add a new element G (and all associated new sets) to M to create a new structure M[G].
The question is :
What is Forcing doing that was impossible to be done during the addition of constants to the model M (so G isn't a constant created during the construction of M)? Is it undertaking some new infinitary process or applying some new expressions or using more constants for each $\exists$xU(x) expression that for some reason couldn't have been applied whilst constructing M ?
So there are a few things going on here.
First of all, the process of constructing a term model that you describe is not actually doable with a computer programm. The problem here is that by Gödels 2nd incompleteness theroem, there is no recursive complete consistent extension of ZFC and so you could never feed a computer the necessary information to complete this process.
Now onto your actual question: Forcing is not really doing anything that the first construction cannot do: It is certainly possible that the termmodel you construct is a forcing extension! That will happen, if the complete consistent extension of ZFC that is used in the construction contains the negation of the so-called "Ground Axiom" (this axiom states that the universe is not a forcing extension of any strictly smaller universe [it is not obvious that this is first order definable, but it turns out it is]).
Maybe it is easier to first think about constructing a term model for the field axioms instead of a model of set theory. As a complete consistent theory you might want to take the deductive closure of the theory of algebraically closed fields of characteristic zero, or whatever entails the most pleasure for you. Lets call the resulting termmodel $K$. Nothing stops you of building the field $K(X)$ of rational functions in $K$ with parameter $X$. Now you have successfully added something which was not build in the construction of $K$. And moreover, you have changed the theory! $K(X)$ is not algebraically closed anymore.
Forcing is doing a similar thing - it adds new information which is in some way consistent with the ground model and possibly changes the theory with it. The good thing is now that it gives us control. In the termmodel construction, you have no control over the theory you feed it. With forcing, you have some control over the theory of the resulting model in a nice way. The map $G\mapsto \operatorname{Th}(M[G])$ is continuous (with the right choice of topologies), where $G$ is a generic for some specified forcing $\mathbb{P}$ in the groundmodel $M$.