Difference between equivalent conductance and equivalent conductivity
Conductance and conductivity are distinct, although related, magnitudes.

Conductance, $G$, is a measure of the total electric current, $I$, that can pass through a conducting element (such as a wire, a space through which a charged beam flows, or an electrolytic solution) when a given voltage, $V$, is applied through it:

$G = \cfrac{I}{V}$

Correspondingly, the IS unit for conductance is the siemens (S), equivalent to an ampere per volt. Since conductance is the inverse of resistance, a siemens is also the inverse of an ohm.

However, in general, conductance does not only depend on the nature of the material; it also depends on the geometry of the conducting element. For instance, the conductance of copper wires of different length and/or different section will be different. Conductivity, $\sigma$, is defined in order to compare the contribution to conductance of materials, isolating it from shape effects.

In general, for a linear electric flow across a conducting element, conductivity increases with cross-section (as more flow is able to pass through simultaneously) and decreases with length (as more resistance is found). Conductivity is thus defined as

$\sigma =  G \cfrac{l}{A}$

which, ideally, should be identical for all conductors made of the same material (in practice, there are a number of effects that can make the relationship between conductance and geometry non-linear, but for macroscopic ohmic conductors it holds fairly well). This allows direct comparison of the electric properties of materials, effect of doping, temperature, etc.

The IS units of conductivity are therefore siemens per metre (S/m).

Note that unlike conductance, which is an extensive quantity, conductivity is an intensive quantity which doesn't depend on the size of the conducting element. According to 19th century naming conventions, conductivity can also be seen as a specific conductance (in the same way specific gravity is an intensive measure for mass, or specific heat is an intensive measure for heat capacity) and indeed that name was widely used in the early days of both electrochemistry and electromagnetism. While in electromagnetism that standard has changed and currently conductivity is the almost universal name for $\sigma$, in electrochemistry that name lingered and conductivity measurements in solutions were reported as "specific conductance" for longer.

Compound the facts that conductivity measures in electrochemistry are usually made with standard-sized cells in which geometry isn't a factor, and that specific values of conductivity (such as molar conductivity or equivalent conductivity) need an additional qualifier, and these magnitudes were often called "molar conductance" or "equivalent conductance" short for "molar specific conductance" or "equivalent specific conductance". However, you'll find that these magnitudes are always reported in units of conductivity (typically $\mathrm{\mu S/cm}$ or $\mathrm{mS/cm}$).

The use of specific conductance to mean conductivity is now considered obsolete; see for instance the IUPAC Gold Book's page for conductivity.