When we first see the shuttle carrying Tom Paris, the registry number is 71325. In the second shot of the same shuttle, the registry number is 1701-D, indicating it belongs to the Enterprise. In the final shot, when the shuttle is flying over the saucer section of Voyager, it has returned to 71325.
After Torres & Kim are rescued on Ocampa, they are beamed directly to Voyager's sick bay, still wearing the clothes given to them by the Ocampan doctors. But when the scene segues back to Voyager's sick bay, Torres is wearing leather civilian clothes and Kim is wearing a uniform. (These clothes had been removed from them on the Caretaker's array.)
At the start of the series the crew is seen using phaser and tricorder models from The Next Generation, however the models for these change to the First Contact designs later in the series. If the ship is lost in the Delta Quadrant with no way of contacting Starfleet, these updated models would not be available.
Tuvok includes the Ocampan city's energy surplus that will "last at least five years" in the evidence he lists in support of the hypothesis that the Caretaker is dying. However, that information was revealed to Kim and Torres by the nurse who tells them how to access the surface. Nobody told the away team that information.
When Janeway first enters Engineering to investigate the warp core fracture, a technician tells her the pressure is 2100 KPa; she reacts as though that is dangerously high. Following the procedure to lower the reaction rate (and thus core pressure), the quoted pressure is 2500 KPa -- higher than before.
Lieutenant Tuvok wears the insignia of a lieutenant commander throughout most of season one in an apparent costuming error. It will not be until the season four episode VOY: "Revulsion" that he is actually promoted to lieutenant commander.