Not quite the same tubes. For the initial Polaris Submarines (which I believe never received Trident C4), to meet stringent requirements to protect the missiles from shock in the original Mk 17 Polaris Launch System, the Polaris missile was held by three pads or "stowage launch adaptors" inside the heavy machined steel launch tube, and the launch tube was in turn held in the submarine mount tube by oil-filled double-acting shock absorbers. The oil-filled shock absorbers were fitted as they retained a "null" which required significant force for them to deviate from, this enabling the precise positioning for the optical alignment required to ready the guidance system.
The opportunity to take advantage of a greater proportion of the volume of the submarine mount tube only came as a result of investigating the use of truck-launched Polaris for the MLF. As the original stowage tube was too heavy, and the the stowage launch adaptors were incapable of supporting the missile when stored horizontally on a truck, Westinghouse came up with a design where the missile would be supported by foam-padded resin-reinforced fibreglass panels with heavy duty zippers up the sides which would be undone after the missile was loaded into the launch tube. This would support the missile regardless of orientation and would simultaneously remove the need for the heavy machined launch tube, the launch tube itself now being cushioned by foam springs. A pair of demonstration trucks were built, but when the MLF died, Westinghouse marketed this to the Navy as a means of increasing the number of missiles per submarine (as it would require smaller submarine mount tubes hence more could be installed within a given length). The Special Projects Office realised it would also ensure that larger missiles could be launched from existing tubes.
This later Mk 21 Polaris Launch System would be introduced from James Madison onwards, but would only have the potential for back fitting larger missiles, primarily replacing the liquid springs suspending the launch tube within the mount tube with polyurethane foam, but retaining the liquid springs on which it rested, as were the existing stowage launch adaptors.
The Mk 21 would modified into the Mk 24 for launching Poseidon via a modular replacement with the thin launch tube and padding replacing the earlier machined launch tube and stowage launch adaptors taking full advantage of Westinghouse's innovations for the truck-based MLF system.
Source: From Polaris to Trident, Graham Spinardi
Lockheed/Smith Miller Hybrid Missile Launch Truck