TL;DR; A mentor does not do hands-on training and can, therefore, be regarded as a Lecturer of a university. A trainer, however, does take on a hands-on approach with tutorials and actual development. A trainer can be regarded as a tutor, keeping the university example in mind.
Both mentors and trainers choose their open dates (within reason) to work to uphold flexibility. There will be milestones, and weekly sprint objectives to be accomplished, and these will dictate how these times are chosen or determined.
A trainer is expected to lead a group of students (trainees) through demonstration and practical assignments. A trainer would not necessarily finish every assigned component for a demonstration throughout the entirety of his assigned course. Still, basic educational practices are expected as you would lead by example.
A mentor is expected to be a guide for students (trainees) for their more theoretical development. Where they would only learn certain lessons through trial and error, ventures like project management, technical writing, and business development among others are essential skills that almost everyone needs at least the basic knowledge of.