Aspirin's more soluble in water:
Soluble. <0.1 g/100 mL at 23 C (from chemfinder)
Acetaminophen is just slightly soluable:
Very slightly soluble. 0.1-0.5 g/100 mL at 22 C (from chemfinder)
Polarity is a measure of the charge distribution of a molecule, which is dependant on the molecule's shape and bond polarity. In otherwords, it depends on where the electrons are concentrated within the structure of the molecule which also depends on the effects of very electronegative atoms.
Polarity is described by its dipole moment where mu=Q*r where Q is the change and r is the distance between the two charges. The units are in the form of charge times a distance and also go by the name debye. So the greater the distance between two charges or the greater the charge between at least two separated charges, the greater the polarity.
Solubility is related to polarity in that polar molecules attract each other much more than attracting non-polar molecules; so the separating of oil and water is not that oil is attracting other oil molecules than that the water is separting from the oil because it wants to maximize electrostatic attractions with other polar molecules, like water.
Here I believe the greater number of highly electronegative oxygen containing groups in the aspirin make it more polar than acetaminophen.