Apostrophe is used most often in poetry to add emotional intensity; the emotional intensity of "Bright Star! Would I Were as Steadfast as Thou Art" comes from Keats' viewing of the planet Venus rising outside his window when he and his friend James Rice stayed on the isle of Wight off the Southern Coast of England. At the time that Keats sighted Venus he was concerned about his troubled love for Fanny Brawne. So, much like Romeo, Keats saw the loveliness of Fanny in the planets.
Thus, in his emotional address to the star, the poet addresses his love. He wishes to be as eternal as the star that keeps watch "with eternal lids apart." And, yet, he does not wish to exist by himself "in lone splendor" like the star. Rather, he yearns to "Awake forever in a sweet unrest" and "Pillowed upon my fair love's breast." It is here that the pardox is created in this sonnet of Keats. For, he cannot be immortal and retain the love for the woman which must change with time although he wishes he could: "Would I were..."
To me, apostrophes are always more powerful than simple descriptions. If Keats said "I wish I were as steadfast as that star up there," it would not seem very passionate. It would seem like he's just sort of walking around and thinking in a casual way.
But when poets use apostrophes the poets seem to me to be very much more passionate than when they don't. It seems like they are so overcome with emotion that they are talking to inanimate objects. So it just makes me feel as it they are much more serious about their emotion.