A puzzle mug is a novelty that is counter-intuitive. It will above all have indefinite holes in the rim, directorial it impossible to brew from in the normal way. Frequently the solution is to cover all the holes in the rim, and then Photo Mugs drink via another cavity in the cupped handle.
A travel mug is a variation on the traditional mug that is better for transporting hot liquids. It may or may not be a vacuum flask, but is habitually well insulated and completely enclosed, with an easily closed opening on the top.