The paper presents the possibility of applying a twin-screw press for juice extraction from apples with different textural properties. The research was carried out with six different varieties; namely, Granny Smith, Modi, Ligol, Lobo, Boscop and Szampion. During the experiment, the following properties were measured: texture properties, pressing yield and polyphenolic content; and ascorbic acid content and antioxidant activity both in raw apples and apple juice. Based on the analysis, three hardness levels of apples can be distinguished, impacting the course of juice pressing in a twin-screw press (low hardness below 30 N, medium hardness 30−50 N and high hardness 50 N). The study showed that only high hardness apples are suitable for pressing on a twin-screw press. The mechanism by which texture properties influence the juice pressing process in the studied press was explained. It was further demonstrated that the hardness of apples has a positive impact on the degree of extraction of polyphenols and ascorbic acid into apple juice. This is caused by the different fracture patterns of tissues with different mechanical properties. High hardness apples (Granny Smith and Modi) were characterized by a considerably greater degree of bioactive substance extraction into juice than medium hardness apples.