We're thinking thrust here, vs lift, though.
So are the pilots. In a jet, pushing the throttle forward increases the thrust from the engine, which always pushes the craft forward; the stick changes the control surfaces which orient the plane. In a helicopter, pulling the collective up increases the pitch of the main rotor, which increases its thrust; moving the cyclic (the stick) directly changes the orientation of that thrust in a hover, allowing flexible hovering motion.
In forward flight, the helicopter and jet fly surprisingly similarly; to go faster but level, in the jet you push (increase thrust), which will alter your pitch as you now have air moving over the wings faster; as I recall this will probably pitch you up a bit, so you'll have to push on the stick to nose over a bit (it's been a while). With the helicopter, you pull (increase thrust), which means you immediately start climbing (airflow is down, so thrust vector is up), so you push on the cyclic a bit to nose over; the flow is now directed backward a bit (thrust vector forward), so the vertical component of lift is cancelled and you're now moving forward faster.
Below effective translational lift, a helicopter and jet fly far different, of course; the helicopter's thrust allows you to hover delicately, while the jet falls out of the sky and everybody dies. But that's a different story.
Or if you have enough thrust to overcome gravity, taking all control-surface lift out of the equation so that it's ENTIRELY a matter of thrust (ie very strong jet pointed straight up, helicopter in a hover), jet pilot pushes forward to go up, helicopter pulls up to go up.
This leads to confusion with the tilt-rotor crowd, depending on which sort of aircraft you flew first...