First of all, they are synonyms and in most (if not all) cases they could be used interchangeably as both imply that something is causing something else's actions. However, they do have slightly different connotations.
When one motivates someone the motivator is typically giving someone a reason (e.g. motive) to do something. For example, money is a reason to work. If I yell at you to exercise you then you have a reason to exercise (fear, intimidation, or a desire to impress) and so you could say that I am motivating you. If you are using motivate then the focus is on a specific reason/motive and it tends to be specific and narrow. You generally wouldn't say a vague statement like "Joe motivates me."
When one inspires someone there isn't necessarily a solid reason. The focus tends to be more on the one doing the inspiring. It can be used when the reason is vague (e.g. if MLK inspires you to do things then there are reasons MLK is creating for you but it would be hard to pin them down) and can be used in vague statements (e.g. "Joe inspires me." is a fine sentence).
I think the word you use depends on what you are trying to communicate. If you want to focus on a specific reason/action relationship use motivate. If you want to keep it vague use inspire.