The modern depiction of the Genie in a Bottle seems to indicate that the lamp is in fact the source of the Genie's power. Without it, he or she is either weakened or turned human. However, the Lamp is also their prison. They must give wishes to whoever rubs their lamp, and cannot resist. They also cannot directly harm their master. See Literal Genie, however, for passive-aggressive means of rebellion; and Jackass Genie for less passive means.
This association of "genie" with "slave" means that we don't, generally, see free genies anymore, and that the intrinsic nature of genie "slavery" can be used as a plot point, as in Disney's Aladdin (You wished to be an all-powerful genie? Now you're stuck in that lamp!). Given the U.S.'s history with slavery, Western depictions of heroes who acquire genies will almost always free them in the end (provided that they're good). What the genie was imprisoned for originally is typically not mentioned.
Another interesting change is how the nature of wishing has become a kind of reality warp. The genie activates some kind of command written into the fabric of existence, and *poof*, the universe is that way. Just as a Genie is slave to the Lamp, the Wish seems to be something more complicated and powerful than they themselves are; they just facilitate its invocation. While they may have some magic tricks they can use for themselves, they cannot consciously use the same powers the Wish facilitates.
Since Genies are usually Shapeshifters, they usually also have Morphic Resonance such as blue skin, Supernatural Gold Eyes, appearing in a puff of smoke or some other feature that distinguishes them in any form. While today's image of a genie is fairly standardized and stereotypically Middle Eastern—a muscular, shirtless man, without legs, in a turban and usually with an ornate Arabic sword—this kind of standard visual preconceptions only seems to have arisen during the twentieth century; earlier depictions of genies by Western artists are very varying
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The djinn were originally spirits of dust devils, hence the term, and almost Always Chaotic Evil in the oldest stories. Whenever you see a dust devil, that is a genie in its natural element. In the desert of the Sahara, or the plains of West Texas, dust devils can be powerful enough to snatch up livestock and small children, and those swirling leaves tend to follow you around, hence the origin of the belief isn't as illogical as you might think.
