High pressure processing (HPP) is an alternative to thermal pasteurization (TP) of beverages that can impart an extended shelf-life while delivering products with “fresh-like” sensory properties. However, studies documenting the sensory changes associated with HPP of fruit juices and their relation to consumer preferences are limited. In this paper, the effect of HPP on sensory quality and consumer perception of orange- and apple juice was investigated using a full experimental design with samples varying in treatment (HPP, TP), storage time (0, 3, 11 weeks for orange juice and 0, 4, 12 weeks for apple juice) and storage temperature (4 °C, 25 °C). Samples were first analyzed by descriptive analysis with a trained panel and subsequently submitted to a consumer liking test (N = 75). Treatment and storage conditions had a significant impact on sensory quality and on consumer perception. HPP could better maintain the sensory quality of both juices. Cold storage was effective in maintaining the product quality over the entire storage period. The main drivers of consumer liking were attributes associated with shorter storage times and/or cold storage (e.g., intensity of orange aroma and flavor in orange juice, apple flavor and aftertaste in apple juice), and to a lesser extent to treatment.