Blackcurrant juices were prepared without enzyme (NEB-juice) and with conventional enzyme-aided berry pressing (EB-juice). Juices were pasteurized and stored at ambient temperature (in light and dark conditions) and at +4 °C for a 1-year period of time. Volatile composition and odor attributes were followed by HS–SPME–GC–MS and a sensory panel, respectively. Volatiles were decreased in the EB-juice, showing 100-fold lower contents of the main terpenoids, e.g., α-pinene, δ-3-carene, limonene, terpinolene, bornyl acetate and β-caryophyllene; whereas esters, such as methyl butanoate and ethyl butanoate, showed 2- and 4-fold lower contents for the EB-juice. Pasteurization produced minor changes in both juices. Volatiles decreased during storage at room temperature while at +4 °C esters and eucalyptol were kept at 100%. The descriptive sensory analysis reported a significant decrease (p < 0.05) of the berry-like odor after 12-month storage at RT. Storage at +4 °C for 12 months did not affect the odor quality.