Back in the early days of tractors, when options like hydraulics and power take-offs were first appearing, they were driven THROUGH the transmission. When you would depress the clutch pedal (or pull the clutch LEVER in the case of hand clutches) you would kill the power going to the pto or hydraulics from the engine. Hence, pto and hydraulics quit "moving".
"Live power" is credited (depending on which version you hear) to either Oliver or Minneapolis Moline. The old "transmission drive had a serious flaw. If you had a pto driven implement on the tractor, and you depressed the clutch to stop moving, the flywheel effect of the implement would continue to "back feed" power to the transmission, even as the power of the engine was no longer being connected to the transmission. That has shoved many an old tractor through the barn door. Live power usually uses a secondary clutch to control input power TO the pto, as well as to isolate "power" being back-fed through the pto shaft to the transmission (by that flywhel effect) The secondary clutch can be a simple 2-stage clutch employing the same clutch pedal as the engine-to-transmission clutch, OR, in some cases, a seperate (USUALLY hydraulic actuated) INDEPENDENT pto clutch. SOME independent pto systems are electrically controlled, via an electromagnetic clutch or solenoid valve.
Live hydraulics are of the same line of thought. Early hydraulic systems had their pumps driven through the transmission. Depress the clutch and loose hydraulic pumping. Live hydraulics have their pump directly driven off the engine, not being effected at all by the clutch. In SOME cases, the hydraulic pump will be driven through the same mechanics as the live PTO, but uneffected by the pto being in or out of gear. In those cases, depressing the second stage of the 2-stage clutch will cease hydraulic function momentarily.
Hope this helps.