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Oxford Dictionaries define the nouns as
So which sentence sounds more natural to a native speaker?
- I studied physics with the intention of becoming a physicist, but I recently felt no vocation to it.
Or
- I studied physics for the purpose of becoming a physicist, but I recently felt no vocation to it.
'With the purpose' implies that the thing you are doing is contributing to your goal. 'With the intention' only implies that you had another goal in mind. The action doesn't have to contribute to it.
So you can say
I went for a walk with the intention of finishing my studies later.
The walk doesn't contribute to the studies, so you could not substitute ' purpose' for 'intention'. If ' purpose' is correct you can usually substitute 'Intention'.
I think there's more here than is mentioned. In the example "I went for a walk with the intention of finishing my studies later", there is a subtle implication that this was not arrived at (finishing of the studies). Purpose is stronger, doesn't weaken the association as much.
Intention - Set of actions,promises or a kind of behavior which can conclude or reveal an ultimate result. Ex. His intention to join politics shows that he will help downtrodden. You can relate it with a Political Manifesto document revealed before public by a party before election.This document have a set of promises and actions based on which you decide what will the particular party will do once it is elected.
Purpose - A clear idea, a plan or a thought. It doesn't have actions initially, it is just a crystal clear plan, that have actions later once we decide something. Ex. Purpose of my life is to utilize the potential I have.
