Rolled-up nanostructures provide new opportunities to control and modify thermal properties. The topology and geometry of nanoscale architectures have a strong influence on phonon spectrum and thermal conductivity. In the theoretical description, rolled-up nanotubes are often approximated with nanotubes constructed from concentric multiwalled nanotubes. Herein, the phonon spectrum is shown not to differ for these two nanotubes if the radius of the nanotube is much larger than the lattice constant. If the contact between layers of the rolled-up or concentric nanotubes is weak, the thermal conductivity is not influenced by the contact, and it can be approximated by the conductivity of the one-layered flat membrane. In the limit of weak contact, the diffusive heat transfer is shown to demonstrate an appreciable difference between rolled-up and concentric nanotubes with large radii. The experimental measurements can be significantly distorted by the effects of the weak interlayer contact.