Illegal = against the law; Illegitimate = outside the law, not "kosher". Something can be illegitimate without being illegal - e.g., a child out of wedlock - but all that's illegal is also illegitimate. In other words, illegal is more clear-cut, and has clearly defined consequences.
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ISIS claims responsibility for Saturday's attack that killed 4 near Cairo (AP)
from the article: After uproar, Netanyahu orders stop to West Bank housing tenders