The boomerang phaser was not introduced until the third season. When Chakotay goes to the Bridge, the crew should be using the older angular phasers from the first and second season, since they have not yet left the Alpha Quadrant.
The Janeway from the past becomes determined to change Voyager's fate to avoid it getting stuck in the Delta Quadrant in the first place. In season 1's "Eye of the Needle," an opportunity to do just that, via having the Romulans from 20 years in the past warn Starfleet about the mission that stranded them there, was presented and roundly dismissed by Janeway.
When Chakotay is holding Captain Janeway hostage in the corridor, he is holding the hypospray at the left side of her neck when the camera is facing their backs. Yet when the camera cuts back to the front he has the hypospray on the right side of her neck. Clearly he could not have moved the hypospray that fast.
The 2371 version of Captain Janeway states that Voyager does not have an Astrometrics lab. However, in Revulsion (1997), Chakotay mentions that the Astrometrics lab hadn't been updated since Voyager left spacedock.
At around 27 minutes, Janeway and Chakotay are in the Captain Proton program. Everything in the program is in black and white except for the panel containing the gel pack, which is displayed in full color. However, just after Janeway says "you'll render it harmless," Chaotica walks toward the panel, and it can be seen at the bottom-right of the screen in black and white. In the next close-up, it's in color again.
The door sensors could not have detected Seven entering Engineering because Seven was in another time-frame.
When Seven of Nine enters Engineering she passes through the temporal barrier. However when the rest of the crew exited through the same doors they aren't shown to have passed through the same temporal barrier.
It's been established that Voyager isn't able to navigate or go to warp without their deflector dish. When it was burned out by the chronokenitic surge they shouldn't have been able to resume course until it's been repaired, which isn't shown on screen at all.
Chakotay tells the Doctor that the inoculation made him immune to the temporal distortions, but technology is still affected. However, Chakotay's com badge and clothes (which were also not inoculated) are able to move through the distortions.
Janeway states near the end that by stopping the anomaly's effects, no one would remember what had happened. So why does it matter what Chakotay tells her?
Chakotay refuses to tell Janeway about his adventures through time, citing the "Temporal Prime Directive." As the anomaly has passed and the splintered timeline has been averted, there is no risk of polluting the timeline, and the Temporal Prime Directive would not apply. There is no reason he couldn't or shouldn't tell her what occurred.
When the past Janeway is going to ask Chakotay one last question in Engineering, the reflection of a boom mic can be seen in the glass near the warp core.
The Doctor tells Janeway that his program has been running for almost three years. Earlier, he told Chakotay that his stardate was 49624; however, that date would only be a little over a year after the Doctor's first activation on stardate 48308.
Chakotay mentions the "Ocampans" a couple of times. The plural form of that race is actually "Ocampa," as established in Caretaker (1995).
Chakotay has been hiding bottles of Antarian cider from Neelix (and from the rest of the crew) for 6+ years; yet, in "State of Flux," he gets angry and punishes Maquis members who stole mushrooms from Neelix so that Seska could make mushroom soup for him.
It's shown that Chakotay is going to Engineering when Ensign Kim says "...the warp core is destabilizing." This is something that Chakotay hasn't done in the past (at least what is shown on screen). It's completely out of character for him to be in Engineering when his place is on the Bridge.
Janeway plans to serve pot roast for her dinner with Chakotay. Chakotay has been established as a strict vegetarian.