Mopping the floor just pushes the dirt into the grouting - think about it.
That's why the grime has accumulated there. The only way to get the grouting clean would have been to run a cloth along all the edges of the tiles after mopping, preferably with a bleach solution, pushing it into the furrows with your fingertip.
The only way to clean the floor would really be to scrub it like in the olden days. I mop mine every week, too, but if I go to pick up an extra bit of dirt immediately after mopping, say, with a piece of loo paper, and the floor is still damp, the paper comes up black and full of dust, fluff and bits. Good job I can't see it all that well without my glasses!