What makes R-MgX a more reactive Grignard reagent than R-Cu?
The difference in electronegativity between copper (1.9) and magnesium (1.3) is the key difference.  

Since copper is more electronegative than magnesium its electronegativity is much closer to that of carbon (2.55).  This results in carbon-copper bonds being less polarized and more covalent than carbon-magnesium bonds.  The electrons in a carbon-copper bond are better shared between the two atoms, whereas the electron distribution in a carbon-magnesium bond places more electron density on carbon.

As a result of this difference in electron density on the carbon atoms the carbon portion of a cuprate is a softer nucleophile, while the carbon portion of a Grignard is a harder nucleophile.  These difference in nucleophilicity produce different reactivity patterns for the two reagents.  For example, the softer cuprate will undergo 1,4 addition to an $\alpha,\beta$-unsaturated carbonyl compound (a softer electrophile), while the Grignard adds directly to the carbonyl (which is a harder electrophile).