I think the Wikipedia page you quote is poorly worded. What on earth does it mean to ‘dissociate into molecules’? Nothing, in my opinion. A better wording is discussed below.
But first, let’s discuss what actually happens when acetic acid is dissolved in water. As you may know, glacial acetic acid consists mainly of $\ce{H3CCOOH}$ molecules that associate to form hydrogen bonding networks. There is practically no ionisation in glacial acetic acid, i.e. the autoprotonation equilibrium $(1)$ is leaning very strongly to the reactants’ side.
$$\ce{2 H3CCOOH <<=> H3CCOOH2+ + H3CCOO-}\tag{1}$$
So before dissolution, we are dealing with molecules of acetic acid. If we add these into water, most of them just stay being molecules; only a small percentage ionises in water according to reaction $(2)$. The acidity constant shown in the equation is a measure of how many molecules are dissociated; it depends on the concentration.
$$\begin{gathered}\ce{H3CCOOH <<=> H3CCOO- + H3O+}\\
K_\mathrm{a} = 10^{-4.76} = 1.74 \times 10^{-5}\end{gathered}\tag{2}$$
So upon dissolution we have:
- most ($> 99~\%$) of the acetic acid molecules remaining acetic acid molecules
- a very small subset deprotonating to form acetate anions
Only the latter are charged compounds and thus only they contribute to the solution’s conductivity. Since they are few in number, conductivity is low.
Getting back to the original quote. It seems to imply that dissolving acetic acid in water ‘turns it’ into molecules; which is wrong: the acetic acid molecules remain the same all through. A better wording would emphasise that they do not change. For example:
Acetic acid is extremely soluble in water, but most of the dissolved compound remains as molecules, rendering it a weak electrolyte.
Acetic acid is extremely soluble in water, but only a small fraction is dissociated into ions, rendering it a weak electrolyte.