Using \bmod in inline math mode seems to add some extra following space, which seems like a bug. Not sure if it's a problem with other binary operators.
This is a consequence of #836, I'm afraid. (Not quite sure why \bmod is implemented with negative space, but it is.) We still don't know how to fix it (or the exact cause).
Hmm...but does it need negative space? If I use \mathbin{\mathrm{mod}} that works perfectly fine: see
https://cdn.rawgit.com/akalin/98bc47f5e54f7529462f12a1d7336858/raw/b71b173e7fdc3459938d7f362574da51eacdad6a/test-0.8.3.html
So I guess that's a workaround for now.
For some reason, LaTeX (latex.ltx) defines \bmod as follows:
\def\bmod{%
\nonscript\mskip-\medmuskip\mkern5mu%
\mathbin{\operator@font mod}\penalty900\mkern5mu%
\nonscript\mskip-\medmuskip}
So KaTeX is mimicking that. Basically, in display/textstyle, we want 5mu minus \medmuskip (which is 4mu plus 2mu minus 1mu skip), while in script/scriptscriptstyle, we want 5mu skip. Given the #836 bug, I think it'd be nice to remove the negative space in \bmod. I've created PR #984 to this effect.
Meanwhile, you can define a macro of "\\bmod": "\\kern1mu\\mathbin{\\mathrm{mod}}}\\kern1mu", assuming you're not using \bmod in subscripts.
Hmm, that \bmod macro seems to have too little space after mod. With
\begin{aligned}
& (a \cdot b) \bmod 257 \\
& (a \cdot b) \mathbin{\mathrm{mod}} 257 \\
& (a \cdot b) \kern1mu\mathbin{\mathrm{mod}}\kern1mu 257
\end{aligned}
I get

\mathbin{\mathrm{mod}} seems closer to \bmod.
Although the 2nd \kern1mu should _add_ space, right? It's weird that putting it in _removes_ space...
I saw that too. Must be a bug of some sort... Should track it down.
The PR has the right space though.