They mean the same thing, but “completely” is the more common and useful word. You can say “the glass is completely full,” but it would sound strange to say “the glass is utterly full.” “Utterly” is usually used for negative things: “the city was utterly destroyed,” “the meeting was an utter disaster.” Even though both mean “100%,” “utterly” is usually used for very extreme situations.

I would say very very similar. Utterly is less common but has more emphasis.