Some paper chasing netted this short article by George Marsaglia, in which he also quotes the article by James Glaisher where the error function was given a name and notation (but with a different normalization). Here's the relevant section of the paper:

In 1871, J.W. Glaisher published an article on definite integrals in which
he comments that while there is scarcely a function that cannot be put
in the form of a definite integral, for the evaluation of those that
cannot be put in the form of a tolerable series we are limited to
combinations of algebraic, circular, logarithmic and exponential—the
elementary or primary functions. ... He writes:

The chief point of importance, therefore, is the choice of the
elementary functions; and this is a work of some difficulty. One function
however, viz. the integral $\int_x^\infty e^{-x^2}\mathrm dx$,
well known for its use in physics, is so obviously suitable for the purpose,
that, with the exception of receiving a name and a fixed notation, it may
almost be said to have already become primary... As it is necessary that
the function should have a name, and as I do not know that any has been
suggested, I propose to call it the *Error-function*, on account of its
earliest and still most important use being in connexion with the theory of
Probability, and notably with the theory of Errors, and to write

$$\int_x^\infty e^{-x^2}\mathrm dx=\mathrm{Erf}(x)$$

Glaisher goes on to demonstrate use of $\mathrm{Erf}$ in the evaluation of a
variety of definite integrals. We still use "error function" and
$\mathrm{Erf}$, but $\mathrm{Erf}$ has become $\mathrm{erf}$, with a change
of limits and a normalizing factor:
$\mathrm{erf}(x)=\frac2{\sqrt{\pi}}\int_0^x e^{-t^2}\mathrm dt$ while Glaisher’s
original $\mathrm{Erf}$ has become
$\mathrm{erfc}(x)=\frac2{\sqrt{\pi}}\int_x^\infty e^{-t^2}\mathrm dt$. The normalizing
factor $\frac2{\sqrt{\pi}}$ that makes $\mathrm{erfc}(0)=1$ was not used in
early editions of the famous “A Course in Modern Analysis” by Whittaker and
Watson. Both were students and later colleagues of Glaisher, as were other
eminences from Cambridge mathematics/physics: Maxwell, Thomson (Lord Kelvin)
Rayleigh, Littlewood, Jeans, Whitehead and Russell. Glaisher had a long and
distinguished career at Cambridge and was editor of *The Quarterly Journal of
Mathematics* for fifty years, from 1878 until his death in 1928.

It is unfortunate that changes from Glaisher’s original $\mathrm{Erf}$:
the switch of limits, names and the standardizing factor, did not apply to
what Glaisher acknowledged was its most important application: the normal
distribution function, and thus
$\frac1{\sqrt{2\pi}}\int e^{-\frac12t^2}\mathrm dt$ did not become the
basic integral form. So those of us interested in its most important
application are stuck with conversions...

...A search of the Internet will show many applications of what we now call
$\mathrm{erf}$ or $\mathrm{erfc}$ to problems of the type that seemed of
more interest to Glaisher and his famous colleagues: integral solutions
of differential equations. These include the telegrapher’s equation,
studied by Lord Kelvin in connection with the Atlantic cable, and Kelvin’s
estimate of the age of the earth (25 million years), based on the solution
of a heat equation for a molten sphere (it was far off because of then
unknown contributions from radioactive decay). More recent Internet mentions
of the use of $\mathrm{erf}$ or $\mathrm{erfc}$ for solving
differential equations include short-circuit power dissipation in
electrical engineering, current as a function of time in a switching diode,
thermal spreading of impedance in electrical components, diffusion of a
unidirectional magnetic field, recovery times of junction diodes and
the Mars Orbiter Laser Altimeter.

On the other hand, for the applications where the error function is to be evaluated at *complex* values (spectroscopy, for instance), probably the more "natural" function to consider is Faddeeva's (or Voigt's) function:

$$w(z)=\exp\left(-z^2\right)\mathrm{erfc}(-iz)$$

there, the normalization factor simplifies most of the formulae in which it is used. In short, I suppose the choice of whether you use the error function or the normal distribution CDF $\Phi$ or the Faddeeva function in your applications is a matter of convenience.