As Benny notes, "omen" is generally used as a noun, while the others are all verbs. (I see that the dictionary says that "omen" can be used as a verb, but I don't recall ever seeing it used that way.)
Besides that, the words all mean pretty much the same thing in English.
There may be some subtle differences in connotation in current usage. Word origins aside, "augur" and "portend" are usually used in a mystical or occult sense, while "bode" and "presage" are generally used in more mundance contexts. Like, someone is more likely to say, "The fulfillment of this ancient prophecy portends the coming of a dark age" then to say that it "presages" it. But they'd say, "A decline in the stock market presages an increase in unemployment rates".
In practice, "bode" is almost only used to say that something "bodes well" or "bodes ill". It's very rare for someone to say, "This bodes a change in the weather" or any other words following "bodes".
But frankly I'm being very subjective here. If someone actually has statistics on usage I'd be amused to hear it.