The aim of the study was to compare the quality of table grapes (cv. Italia) held on the vine compared to grapes stored in cold rooms with or without modified-atmosphere packaging (MAP). The grapes were harvested from 12 plants in 2 vineyards in the same area, differing for the age of the plant. Four- and a fourteen-year-old vines were cultivated with the “Apulia tendone” system. After the first harvest, grapes were divided into small clusters and used for storage treatments in air and in MAP. Samples of 400 g were packaged in polypropylene (PP) trays sealed with a polypropylene/polyamide (PP/PA) film with 20% CO2 in air. MAP and control samples were then stored in the same cold room at 0 °C. Initially and after 8, 21, and 28 days, grapes stored in air and MAP were compared to fresh harvested grapes, stored on the plants. Quality attributes included color, texture, maturity index, phenols, antioxidant activity, sugars, organic acids, sensory parameters, and volatile compounds. The results obtained demonstrated that grapes held on the plant and in MAP showed better quality in terms of appearance scores compared to grapes stored in air. In particular, the application of high CO2 contributed to reduce the deterioration rate of the clusters, minimizing weight loss, and delaying degradation processes, and this particularly for grapes from the 14-year-old vine, where grapes held on the plant degraded faster than grapes in the younger vines. Most volatile compounds did not change their concentration with the storage treatment, except for ethyl acetate and ethanol, which increased in MAP at the end of storage, and to some compound responsible for green odor. In conclusion, keeping the grapes on the plant can be considered a good agronomic practice to preserve the quality, whereas MAP can be applied to better maintain postharvest quality of the product throughout storage and distribution.