Extremely. Even though the Roman government and body politic was based in Rome, the provinces were extremely important for different reasons. Food was a big area. To meet the food needs of a large, metropolitan city such as Rome, a large agricultural effort was necessary. One of these needs was the grain dole used to keep the poor and destitute of Rome fed and unlikely to rebel. This is why Egypt was such an important Roman province, due to their great agricultural yields. Furthermore, a large portion of the Roman army was drawn not from the senators and patricians in Rome, although they did often serve on the military, especially in the Republic, but from the provinces. The necessity of the provinces can be nicely seen when, during the second Punic war, Rome either completely lost or lost contact with many of its Provinces, although at this point there weren't as many of them as would later come to be. Anyways, depriving Rome of just its Italian provinces nearly brought it to its knees.