Let's suppose someone has a goal and they work on achieving that goal. Would it be idiomatic to say:
He is purposeful.
He is motivated.
Do these two phrases mean the same? If they don't, could you please explain the difference?
Either of those phrases would be idiomatic, and both could apply in that situation, but they don't mean something.
Purposeful means "full of purpose" (more or less), or more naturally that something is done with purpose. A purposeful action is an action taken to achieve a (useful) purpose. It's also the name of a particular leadership technique, but that is irrelevant. If a person is purposeful, then the general trend of their choices in life is to act towards a purpose; the meaning has developed, however, to characterise the determination and persistence that people tend to have when they are acting towards a purpose that is important to them. Thus, saying he is purposeful will generally be taken to mean that has a goal and works towards it with determination.
Motivated means "having motivation" (except when it's motivated by, in which case it is describing the specific motivation). Like purposeful it has then developed to describe the characteristics of strongly motivated people - hard working and determined. Thus, he is motivated will tend to mean that someone is driven, hard working, determined, that there is some thing or things that cause them to work hard.
Thus, the difference in meaning is subtle, but noticeable.