TL;DR: column generation on the dual problem is 100% equivalent to cutting plane on the primal problem.
Equivalence between primal and dual form
Consider the (primal) LP problem
\begin{align}
(P) \ \ \ \min_{x \in \mathbb{R}^{n}} \ \ \ & c^{T}x\\
\text{s.t.} \ \ \
& \sum_{j=1}^{n} a_{i, j} x_{j} \geq b_{i}, & i = 1, ..., M,
\end{align}
with $M$ very large (thereby motivating a cutting-plane approach).
The corresponding dual problem is
\begin{align}
(D) \ \ \ \max_{y \in \mathbb{R}^{M}} \ \ \ & b^{T}y\\
\text{s.t.} \ \ \
& \sum_{i=1}^{M} a_{i, j} y_{i} = c_{j}, & j = 1, ..., n,\\
& y \geq 0.
\end{align}
It is straightforward to see that adding the constraint $a_{i, .}^{T} x \geq b_{i}$ in the primal is equivalent to adding the corresponding variable $y_{i}$ (and associated column) in the dual.
Thus, any cutting plane oracle for the primal is a pricing oracle for the dual, i.e., whether you look for violated cuts in the primal, or positive reduced cost variables in dual, the procedure is the same.
(to view this, start from the complementarity slackness condition and show that the reduced cost of variable $y_{i}$ is the slack of constraint $a_{i}^{T}x \geq b_{i}$)
Consequently, there is no theoretical argument for using one form over the other: the differences are purely computational.
Computational & implementation differences
Should one use the primal or dual form? The only universally valid answer can be: "it depends".
Primal vs dual simplex
In a cutting plane-based approach, you iteratively add constraint to your LP formulation.
This is best done with a dual simplex algorithm (just like in branch-and-cut).
It happens that (i) dual simplex tends to be more efficient than primal simplex, and that (ii) most LP solvers provide a nice API for doing this constraint generation (via so-called "lazy constraints").
In a column-generation approach, you iteratively add variables to your LP formulation; this is best done using the primal simplex.
Furthermore, column generation requires dual information (recall that we're solving $D$, so we're talking about dual solutions for $D$): if you're using simplex that's OK, but if you have something more exotic this information may not always be available.
Unfortunately, most LP solvers do not have any column-generation API. This means you must manually solve the current LP, get the current solution, call your pricing oracle, add new variables if any, and re-solve. This means more work (and more opportunities for bugs).
Overall: if you can try both, great. If not, I'd recommend sticking with a primal cutting plane-based approach.
Primal-dual column generation
One way to implicitly use both primal and dual forms is the so-called primal-dual column generation method (this paper also provides some background on column-generation in the LP case).
It has been found to outperform simplex-based implementations on some large-scale problems.
However, software tools for doing so are very scarce: this is the only one I know of.
It's not too hard to implement yourself, but it requires some familiarity with interior-point algorithms.
For completeness, a closely related method is the Analytic Center Cutting-Plane Method (ACCPM), which is one of the most effective approach for, e.g., multi-commodity network flow problems.
One reason why the primal-dual column generation is so effective is because it implicitly provides a stabilization effect, which I discuss next.
Acceleration & stabilization techniques
So-called stabilization techniques help speedup column-generation (or, equivalently, cutting-plane) algorithms.
From a primal (cutting-plane) perspective, the idea is that the point to separate should not move "too much" between successive iterations.
Most of the time, there is little theoretical justification (like a proof of faster convergence) for them, but they generally work well.
For background:
