You found another here
All my time was on Tridents (Ohio Class). Your post makes me want to write a lot so TLDR warning!!!
First off, I never recalled us being called pilots, ever. You were either a helmsman and/or planesman or unique watch called the Maneuvering Watch Helmsman. All helmsman and planesman are enlisted. And lets just say you are talking the lowest enlisted ranks E1..E3 and maybe a 3rd class Petty Officer (E4) once in a while.
On a submarine, if you are enlisted and not a Nuke or other 6yr Tech school program your bubblehead life starts by washing dishes (Mess Cook). Those are like 37hr long days and expect to do at least 90 days (sucked balls). While you do that you should be rather motivated to earn your Dolphins and qualifying for a watch. The easiest watch to qualify for is the Helmsman/Planesman. This is a fair amount of time sitting at the stations with a qualified helmsman/planesman right behind you. At all times the Diving Officer Of The Watch (DIVE) is right behind the rest of us and the Chief of the Watch (COW) is right to our left. There is also studying manuals and full motion simulator time.
Ironically, the only time officers actually drove the boat was during their qualifications for either DIVE or Conning Officer(CON)+Officer-Of-The-Deck(DECK). It seemed they just needed to be able to do what they will later be ordering the helm/planes to do.
First off, on all the boats with stern planes (horizontal stabilizer with elevator), rudder and either bow planes or fairwater planes (up on the conning tower) there are no pedals. Yep, turning the boat is done by turning the yoke. So that is probably the fundamental control difference. Obviously the thrust is from the rear and things are moving way slower.
@Nub_Pilot indicated a joystick like control. I believe that control system is more like what a pilot would experience however the actual mechanics of it are not at all like an airplane and probably classified (at least it was when I saw how it was going to made).
So back to the standard setup, for turning: You turn the yoke left the rudder swings left. Unlike an airplane and a lot like a ship as you near the final new bearing you actually need to swing the rudder the other way to check the turn. That is one of the things you need to learn for different 'Bells' (speeds) and different turn rates. So you are not banking the sub to turn. New and faster subs will indeed bank but due to other forces but not due to actual control surfaces. The best example of this is that the bow planes or fairwater planes both move the amount and same direction so no like ailerons.
The maneuvering watch helm job is more stressful. Endless fine course corrections and you enter shallow water, tides and ultimately up the pier and you can never see anything. You pin the stern planes in a partial up position to keep the rudder and screw pulled underwater. Definitely not as fast on the surface. The stern planes seat is empty.
The helm has to do 2 things underwater. He controls turns and fine depth control (up/down). So his job is the most pilot like in that its a bit more task overloaded, especially at periscope depth in heavy seas (hated that, not easy to control 18,000tons in 1ft increments with 15ft waves) or some ops with lots of turns and fine depth control.
The stern planes station mainly does just one thing. Control the ships angle. At low speeds its a pretty quiet boring job. At the highest speeds and a screw-up and you can transition hundreds of feet in a very short time. You can only go down so far and not kill everyone. At high speeds this station also maintains depth control as using the planes up front will majorily cavitate (they won't break off, done it many times) but they make noise that can heard forever away. And it feels like the entire ship will rip apart.
The stern planes biggest emergency if a hydraulic failure, especially in the down position.
At no time does these "pilots" directly control speed. The Conning Officer (CON) or Officer Of The Deck who is also the con tells you what speed he wants. The helm enters it on the engine order telegraph. Then engineering (nukies) do nuclear/steam stuff and make it go faster, slower forward, reverse, etc.
The most fun is high speed runs and being both helm and front planes. Lots at stake, especially when operating very close to other big things above or below water.
I'd love to tour a Seawolf or Virginia class. Probably a bit more like a pilot. But I would say very little skill of helming a submarine helped in becoming a pilot. The corollary would a single officer (PIC) sitting in the seat and doing everything himself including navigation, engine control and monitoring.
