"Goodbye" is the accepted spelling according to most dictionaries. The hyphed version is older, but still in wide usage. It should not be two separate words.

They're the same.

Write "goodbye." "Goodbye" and "good-bye" are the same. "Good bye" is incorrect. "Good-bye" appears in old books but it's out of date. Sometimes spellings change slowly over time. There has been a trend to uses fewer and fewer hyphens and to spell more and more compound words as single words, without hyphens. The general rule is simple: check a dictionary. An interesting example is the word "cooperate." That's the current, correct spelling. A problem is that it looks as if you'd pronounce "coop-" as one syllable, like the word "coop." In very old books they put a dieresis over the second "o," "coöperate," to show that you pronounce it as two syllables, one of the very rare uses of a diacritical mark in English. Then they began spelling it "co-operate." Finally they dropped the hyphen, and began spelling it as "cooperate," and you just have to know how it is pronounced.

Both Peachey and Bud are right. "Goodbye" is the most common, I think.