It's because at 1M concentration the two cells would have an equal potential difference and when you link up two 1M half-cells of the same electrode and shit together it will cancel out to give you 0 potential difference so charge (and hence current) will not flow between the two.
However, when you link a 1M half-cell up with a (lets say) 4M half-cell, the 4M does not have a standard potential difference (e naught) value given by the electrochemical series, it is quite different due to not being in Standard Lab Conditions (SLC), which is what the electrochemical series in our data books are determined from.
So since it is a non-standard cell, we can deduce that the 4M half-cell will have a different potential compared to the 1M half-cell, and when we link them up via a wire, the potential differences will add to give a non-zero difference across the whole cell and thus current should flow.