I found at least one dictionary that listed meager as a definition of paltry, suggesting that the two words could be used interchangeably in some contexts, particularly if you are talking about an amount.
However, if you look up both these words on Wordnik, you see some subtle differences in their meaning.
Paltry suggests something is contemptible, unimportant, and insignificant; while meager suggests something is deficient or scant.
When applied to a salary, then, a paltry salary wouldn't necessarily be lower than a meager salary, but it might suggest that the wages were insultingly low.
Although Wordnik lists both words as "equivalents" (you see that at the bottom of the page), the synonym list varies rather significantly. Synonyms for paltry include: worthless, contemptible, despicable; synonyms for paltry include: barren, gaunt, and impoverished.
I strongly suggest looking at both pages on Wordnik, as those two web pages can provide much more information than my meager contribution.
Because there's a lot of overlap between the two words, in many contexts they're interchangeable. But in some specific idiomatic forms (i.e. - "It's a fair cop", "We fight a just war") only one is used.
To the extent that there's a semantic difference, I would say that a fair settlement, for example, is one where the parties directly involved feel a satisfactory compromise has been reached. But to me at least, a just settlement is one that meets external criteria of correctitude (opinions of outsiders, formal moral/legal codes, etc.).
Best Answer
I'm happy is a simple sentence that expresses the state of mind of the person (Subject) at that particular instance. There will be a specific reason behind the happiness expressed on the sentence.
For example: I am happy to know that you got state first in board exams.
I have a happy heart is poetic self explanation of his character by the Subject (I). This expresses that the person is easy going and he doesn't have any stress or hard feelings dumped on his mind. A state of peace.
For example: I have a happy heart despite the serious problems in my life.