抜ける and 出る: emerging through a space and leaving a place
Compare emerging from a forest with leaving a room, separating a tunnel exit from arrival at the next destination.
Passing through a space and emerging
抜ける, read ぬける, here means passing through a space and emerging. 森を抜けると、湖が見えました。 says that a lake came into view when someone emerged from the forest.
細い路地を抜けて広場に出ました。 says that someone passed through a narrow alley and emerged into the square. 路地 is the route, and 広場 is the place reached.
Leaving a place for the outside
出る, read でる, can describe leaving a place. 部屋を出ました。 says that someone left the room. It does not establish a journey across the entire room from end to end.
建物を出てから雨に気づきました。 says that someone noticed the rain after leaving the building. The noticing happened after the exit.
A route through and a later destination
トンネルを抜ける and トンネルを出る can overlap for emerging from a tunnel. 抜ける emphasizes the route through it, while 出る emphasizes the exit.
トンネルを抜けましたが、港にはまだ着いていません。 says that someone emerged from the tunnel but has not yet reached the harbor. The two arrival points have different completion states.
Which movement is complete in トンネルを抜けましたが、港にはまだ着いていません。?
Show the answer and why
Emerging from the tunnel is complete. The sentence explicitly says that arrival at the harbor has not yet occurred.