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, 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". 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").