IMHO, they are all different types of BCDs and all perform the exact same functions.
A travel BCD is simply any sort of BCD that is made with a focus on light weight and reduced bulk. They are gaining popularity due to increasing airline baggage weight restrictions/costs.
A BP&W (Backplate and Wing) is a modular design of BCD, based around a solid backplate (steel, aluminum or nylon) to which you attach your choice of wing (bladder) and harness. You can also attach other functional elements, such as integrated weighting and pockets, if required.
Obviously, the modular design is a benefit in itself; as it allows the diver to specifically customize their rig for their own requirements. As such, it is very popular with divers who prefer a minimalist approach to their diving (more streamlined, less bulky).
Another issue is fitting. A BP&W can be put together and adjusted to create a perfect fit for the individual diver. This makes it a very good for 'odd-sized' divers. However, most BP&W users (of any size) will testify that they tend to fit much better than the standard jacket BCD. One reason for this is that the variable volume bladder (the wing) is mounted behind the diver and does not envelop them like a jacket BCD does. This means that the fit of the BCD does not vary when the diver adds or removes air to the bladder.
Another benefit of the modular BP&W approach is that it can easily be adapted or upgraded over time. This creates some element of 'future proofing' as the diver's needs and requirements alter over time and experience. The BP&W can be upgraded for double tanks and can easily form the basis for a full technical or cave diving configuration.
Most BP&W divers will also state that the BP&W enables more instinctive development of good horizontal trim and buoyancy. This is because the backplate itself spreads a proportion of the diver's weight over the length of their torso and, in addition, the bladder (wing) itself allows an even spread of air/buoyancy. In contrast, the use of a weightbelt and jacket bcd tends to create a 'see-saw' buoyancy problem, where a weightbelt (below the 'fulcrum/balance point of a horizontally trimmed diver - lower torso/hips) pulls down the diver's lower body, whilst the compensating air (above the fulcrum/balance point of a horizontally trimmed diver - upper torso) raises their upper body.
BP&Ws tend to have less material and bulk than jacket BCDs. This should logically increase their stream-lining through the water. A more stream-lined diver will encounter less water resistance and therefore expend less energy on a dive. Less energy/effort expended should also reflect in a lower air consumption. The streamlined nature of a BP&W is also highly popular with divers who operate in restrictively sized environments; such as caves or inside shipwrecks.