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").


  [1]: http://intlpress.com/JDG/archive/pdf/1969/3-1&2-1.pdf
  [2]: http://front.math.ucdavis.edu/0812.1407