Someone who always looks for the "better" and more ideal places (to live), jobs, things to buy that fit his taste/mood etc. He is always looking for better because he always feels it's "not enough".
Although Maulik V. is right in stating that what you describe is human behaviour, not all humans display the behaviour in such a way that it deserves special attention. For the cases where the behaviour is note-worthy (and potentially annoying), you could use opportunist: it describes someone who will see and seize any opportunity to advance their own position. As an adjective, you can describe the person as opportunistic.
As for the neutral description of the behaviour, as you mentioned in the question already, that person is looking for greener pastures. An expression that also fits is the grass is always greener on the other side of the fence, which describes the tendency that humans display to always assume life is better somewhere else (in a literal or figurative way).
This is basically a person who always feels "not enough". This is exactly the opposite of a contented person, someone who is content with his or her life. So, I'd like to suggest this for such a person: a discontented person.
A discontented person is someone who feels unhappy with his or her life. This might be a result of several possible reasons, such as they feel like they are prevented from achieving something (they're frustrated), they might feel that they're treated unfairly (they're resentful), or they might just simply want more (they're insatiable).
A few words with a similar meaning to insatiable are greedy, voracious, and esurient.
As an aside, I googled for these words to see if anything interesting might come up, and I found this interesting essay:
"In subservience to the internal demands of this system [modern commercial culture], the elementary need for material sustenance, for the basic requisites of life, becomes blown up into an insatiable urge for status, power, and luxury. The masters of commerce strive to create in us a perpetual discontent, to induce feelings of inadequacy, to stir up the need to purchase more. As a result, envy and resentment replace contentment; titillation replaces satisfaction; prestige value eclipses life value. The one word to be banished from the dictionaries is “enough.” For the corporate-based economy to flourish there must never be enough, but always a thirst for more: for the bigger, faster, and better; for novelty and variety."
–– Facing the Future: Four Essays on the Social Relevance of Buddhism, by Bhikkhu Bodhi