I figured my comments are getting too long so I should put some of them here, even though I'm using a different topology on $\pi_1$ (often it is the same, but certainly not always) and a possibly different pro-group (which I can't tell as I'm not familiar with toposes).

The relationship goes in 2 steps. First the fundamental pro-group can be compared with the Cech fundamental group endowed with the inverse limit topology. In particular, as noticed by [Atiyah and G. Segal][1], they contain precisely the same information as long as the fundamental pro-group is Mittag-Leffler. For other results in this direction see lemma 3.4 in ["Steenrod homotopy"][2]. The topological homomorphism between $\pi_1(X)$ (with topology as in my comment) and the topological Cech fundamental pro-group is discussed in theorem 6.1, section 5 and elsewhere in "Steenrod homotopy". 

To summarize the relationship very roughly, $\pi_1(X)$ topologized as in my comment retains much of the inverse limit of the fundamental pro-group, discards all of its derived limit, but instead gets something of the derived limit of the second homotopy pro-group (exactly how much is still a subject of ongoing research, see Theorem 6.5 and remark to corollary 8.8 in "Steenrod homotopy"). Still this is not all that it contains (cf. example 5.7 in "Steenrod homotopy").

**Added later:** Addressing Mike's comment, let me elaborate on the choice of topology on $\pi_1$.

The Mathoverflow [thread][3] on the quotient topology on $\pi_1$, and particularly Andrew Stacey's answer there, make it clear that the only reason that this topology is not compatible with multiplication is that the product of two quotient maps need not be a quotient map. But wait, the product of two quotient maps is a quotient map in the category of uniform spaces and uniformly continuous maps! See Isbell's "Uniform spaces" (1964), Exercise III.8(c). [In fact][4], the product of any (possibly infinite) collection of quotient maps is a quotient map in the uniform category. (Quotient maps are defined in any concrete category over the category of sets, as explained e.g. in [The Joy of Cats][5].)

This means that the *topology of the quotient uniformity* on $\pi_1(X)$ makes it into a topological group. By this topology I mean the following. If $X$ is compact, it carries a unique uniformity, and then the space of continuous (=uniformly continuous) maps $(S^1,pt)\to (X,pt)$ is a uniform space (endowed with the uniformity of uniform conergence; if $X$ is metrizable by a metric $d$, this uniformity is metrizable by the metric $D(f,g)=\sup\limits_{x\in X}\ d(f(x),g(x))$.)
If $X$ is not compact, then we need to fix some uniformity on it and the above works. To what extent the resulting topology on $\pi_1(X)$ depends on the choice of uniformity is a good question.

This said, I no longer see any reason to discuss the quotient topology on $\pi_1(X)$. That was just obviously a wrong topology. The interesting question is, does the topology of the quotient uniformity coincide with the "inverse limit" topology, at least when $X$ is compact?


  [1]: http://intlpress.com/JDG/archive/pdf/1969/3-1&2-1.pdf
  [2]: http://front.math.ucdavis.edu/0812.1407
  [3]: http://mathoverflow.net/questions/26680/fundamental-group-as-topological-group
  [4]: http://dx.doi.org/10.1016/0016-660X(78)90033-8
  [5]: http://katmat.math.uni-bremen.de/acc/acc.pdf