In some papers the term 'Transmission Loss' is used. What is the difference between Confinement loss and Transmission loss in a photonic bandgap fiber? Are the same?
Transmission loss is the loss due to the finite absorption of the material it selves, confinement loss is caused by putting your mode in a small space which causes the mode to also propagate partially out of you fiber (meaning not in the center of the fiber). A general example to describe confinement loss is a wave confined in a waveguide slap. It easy to solve an you will directly see where the modes are (and be able to calculate the confinement factor). Yet these are the definitions for fibers with a core and cladding, so it can be that the definitions could be a little different to the photonic bandgap,where I do not have any experience with.
Confinement losses are the losses arising from the leaky nature of the modes and the non-perfect structure of the PCF fiber. Then, depending on the wavelength, number of holes rings, and hole size, modes will be guided with a structure dependent loss. You can find some papers talking about that as in "Confinement losses in microstructured optical fibers", Optics Letters Vol. 26, Issue 21, pp. 1660-1662 (2001). Transmission losses are mainly due to material absorptions, as already said by Stephan above. I hope that it clarifies a bit your doubts.