* The speaker of parliament is responsible for nominating and dismissing the prime minister (who appoints the prime minister? the parliament as a whole appoint the prime minister? does the king get to, but is compelled to appoint whoever the speaker nominates? does wikipedia mean "appoint" when it says "nominate"?)
* The king is not part of parliament, and bills do not need royal assent to become law.
* The king is not part of an executive council like the UK privy council. Formerly there was a Council of State.
However the king does have royal immunity from criminal prosecution. If the Swedish King killed you, they would probably have a crisis over whether the absolute immunity the king possesses is in fact not absolute.
So the king is clearly a special person in Sweden, but it does seem they are almost powerless according to the constitution, unlike say the British queen.