You already have two helpful answers related to general aspects of Eichler--Shimura isomorphisms in a $p$-adic context. Here is an answer that more directly addresses your original question.
I will begin by recalling/stating some facts on the $p$-adic modular form side:
Fix a tame (i.e. prime-to-$p$) level $N$, and let $\mathbb T(N)$ be the completed Hecke algebra generated by the $S_{\ell}$ and $T_{\ell}$ for $\ell \nmid N p$ acting on Katz's space $V(N)$ of generalized $p$-adic modular fuctions of level $N$. Fix a maximal ideal $\mathfrak m$ in $\mathbb T(N)$, and let $V(N)_{\mathfrak m}$ be the localization of Katz's space at the maximal ideal $\mathfrak m$.
In fact, in order to deal sensibly with oldforms at $N$, I find it helpful to do the
following: fix the finite set of primes $\ell_1,\ldots,\ell_n$ dividing $N$, and take
a direct limit of $V(N)_{\mathfrak m}$ as $N$ ranges over all levels divisible by
just $\ell_1,\ldots,\ell_n$. Note that $\mathbb T_{\mathfrak m}$ may grow as $N$ increases (if $N$ divides $N'$ then $\mathbb T(N)\_{\mathfrak m}$ is a quotient of $\mathbb
T(N')\_{\mathfrak m}$), but eventually stabilizes (even thought the $V(N)_{\mathfrak m}$ don't stabilize), because if $\rho$ is any lift of
the Galois representation $\overline{\rho}$ attached to $\mathfrak m$ then the difference between the prime-to-$p$ conductor of $\rho$ and $\overline{\rho}$ is bounded.
So now let $V_{\mathfrak m}$ be this direct limit of $V(N)\_{\mathfrak m}$, and let $\mathbb T_{\mathfrak m}$ be the Katz Hecke algebra acting on it. Note that $V_{\mathfrak m}$ is a smooth representation of the product $\prod_{i = 1}^n GL_2(\mathbb Q_{\ell_i}).$
Note also that there is an action of $U_p$ on $V_{\mathfrak m}$. (Let me reiterate that I did not include $U_p$ in my Hecke algebra $\mathbb T_{\mathfrak m}$!)
Now completed cohomology:
If we take completed cohomology at tame level $N$, we get a $p$-adically complete
$\mathbb Z_p$-module $\widetilde{H}^1(N)$, with an action of $\mathbb T_{\mathfrak m}$,
as well as of $G_{\mathbb Q}$ and $GL_2(\mathbb Q_p)$. We can complete it at $\mathfrak m$
to get $\widetilde{H}^1(N)\_{\mathfrak m}$.
If we then take the direct limit over all $N$
which are divisible exactly by $\ell_1,\ldots,\ell_n$, we get a module I'll denote
$\widetilde{H}^1_{\mathfrak m}$, which has an action of $\mathbb T_{\mathfrak m}$, of $G_{\mathbb Q}$
and $GL_2(\mathbb Q_p)$, and also of the product $\prod_{i=1}^n GL_2(\mathbb Q_{\ell_i})$.
Now suppose that $\overline{\rho}$ satisfies some technical conditions, irreducibility
being the most significant one. (The precise conditions are in the local-global compatibility paper that you mention. Note also that the irreducibility assumption eliminates the distinction between cohomology and cohomology with compact support, and --- more or
less equivalently --- the distinction between working on closed vs. open modular curves.)
Then you can show that there is an isomorphism of $\mathbb T_{\mathfrak m}[G_{\mathbb Q}\times GL_2(\mathbb Q_p) \times \prod_{i = 1}^n GL_2(\mathbb Q_{\ell_i})]$-modules
$$\widetilde{H}^1\_{\mathfrak m} = \rho^u \otimes_{\mathbb T\_{\mathfrak m}} \pi^u
\hat{\otimes}_{\mathbb T\_{\mathfrak m}} V\_{\mathfrak m}^{U_p =0}.
$$
Here $\rho^u$ is the univeral modular deformation of $\overline{\rho}$ over $\mathbb T_{\mathfrak m}$, $\pi^u$ is a representation of $GL_2(\mathbb Q_p)$ on an orthonormalizable $\mathbb T_{\mathfrak m}$-Banach module constructed from $\rho^u_{| G_{\mathbb Q_p}}$ via the $p$-adic local Langlands, and $V_{\mathfrak m}^{U_p = 0}$ is, as indicated, the kernel
of $U_p$ on $V\_{\mathfrak m}$. Also, the completed tensor product $\hat{\otimes}$ has to be suitably interpreted. (One should cut back to a fixed tame level $N$, then form the completed tensor product, and then take a direct limit over all $N$.)
So this is a kind of $p$-adic Eichler--Shimura isomorphism relating $p$-adically completed cohomology and $p$-adic modular forms.
It brings out the difference between the two sides quite clearly: on the completed cohomology side, we have a Galois action, which is encoded in the appearance of $\rho^u$.
(This reflects the classical fact that every cuspform appears "twice" in cohomology.)
Also, completed cohomology has an action of $GL_2(\mathbb Q_p)$, while $p$-adic modular forms just have an action of $U_p$.
So to compare the two, we have to first get rid of the $U_p$-action on $p$-adic modular forms (which we do by passing to $U_p = 0$), and then add in a $GL_2(\mathbb Q_p)$-action, which we do by tensoring with $\pi^u$.
Note also that this isomorphism is not canonical. In this sense, it is analogous
to looking at classical cohomology of modular forms with say $\mathbb Q$-coefficients,
and modular forms with $\mathbb Q$-coefficients. These will be isomorphic as Hecke modules
--- up to the issue of cuspforms appearing twice in cohomology --- but not canonically so. In order to make the Eichler--Shimura isomorphism canonical, one has to extend scalars to an appropriate period ring. Whether this is possible with completed cohomology I'm not sure about at the moment.
One more remark: trading in a $U_p$-action for a $GL_2(\mathbb Q_p)$-action is a fairly significant upgrading of structure, and this is why completed cohomology provides a useful tool for proving modularity theorems for Galois representations, over and above the already-existing theories of $p$-adic modular forms and $p$-adic modular symbols.
Added: You asked about motivation. The original motivation for defining completed cohomology was to construct eigenvarieties. Later it became clear that it was an important object in its own right, providing a global counterpart to the representations of $p$-adic groups that were beginning to appear as part of $p$-adic local Langlands. E.g. the theorem that locally algebraic vectors in cohomology are classical was first proved as an ingredient in the proof of an analogue of Coleman's "small slope implies classical" result for the eigenvariety constructed from completed cohomology. Only later was it realized that this could be combined with a local-global compatiblity result to prove modularity theorems for Galois representations.
Note that the rough relation with $p$-adic modular forms, namely that one gets the same Hecke algebra via either approach, was clear from the beginning, even though the more precise Eichler--Shimura-like statement above was not. Since eigenvarieties (as their name indicates) only care about Hecke eigenvalues, this meant that completed cohomology was good enough for constructing them.
