There is a slight difference, but a difference nevertheless. In my mind, the difference lies in who is the specific agent of action. In the notion of motivation, I think of the idea of someone galvanizing another into action or continuation of action. There driving force is external, as there is one person who is pushing the attempts of another. In this light, there is a direct exertion of force from one to another. The idea of inspiration is more individualistic and centered on self. In this conception, there is a spark that has been lit and cultivated, and then the individual uses this image or mental projection to drive their own efforts and attempts. In this concept, I see that the relationship is more internal and more animated by self. Sometimes, I can see motivation as being morphed into inspiration, the fact that someone stands over can be inspiring.
Motiviating means to propel people to act, to impel them to move, to provide incentive. To motivate seems to suggest that people will receive a reward from the performance that they do for their motivator.
Inspiring has to do with arousing, influencing, producing, or animating (making come alive). This to me, sounds like gaining an intrinsic or internal value from the speaker that drives one to work or perform from within. Motivation sounds more like an external or extrinsic force must act upon a person to make the work or performance occur.
A coach would be happy with inspiration as opposed to motivation because once you get inside someone, they continue to motivate themselves.
The English word 'inspire' comes from the L. inspirare meaning 'blow into.'
The English word 'motivate' comes from the French 'motiver' meaning to 'stimulate toward action.'
The etymology of these two words should thus make the difference in meaning between these two words clear.
To motivate people would imply that a reward or a bribe is offered to make people do something. The pressure is external to achieve some practical goal.
To inspire people would imply that you persuade people to do something merely for the joy or satisfaction of doing something good. The desire for success springs from within a person to achieve his aim and goal in life.