Why are potential differences equal across two capacitors in series, but charge on each capacitor is not?
This is based on a false premise. There is no rule that says that "potential differences are equal across two capacitors in series".
In a parallel combination of capacitors potential difference across each capacitor is same but each capacitor will store different charge. Why is this true?
In a lumped circuit model, any two devices in parallel must have the same potential across them. This is because of Kirchoff's voltage law (KVL) which says that the net potential drop around any loop in a planar circuit is zero. To put it in more basic terms, if the potential at point 'A' is $V_a$ and the potential at point 'B' is $V_b$, then the potential difference between points A and B is $V_a - V_b$ no matter what path you take through the circuit between those points.
As arvindpujari's answer points out, since the potential differences are equal across the two parallel capacitors, this means that if the capacitance values aren't equal then they must have different charges since a linear capacitor is defined by the equation $V=Q/C$.
Also, in series, why are Potential difference across each capacitor different, while charge is the same?
Imagine you start with two capacitors in series with no charge ($V=0$). Then start driving a current through them. The same current will flow through both capacitors (that's what it means for two elements to be in series), so the same charge will accumulate on each capacitor's plates.
Since they have equal charge, if the capacitance values are not the same, then the potentials must also not be the same.