In my mind I do not see a difference between a king and a kalipha, a king is usually chosen by the people, and a kalipha is also chosen by the people, I do not really see a difference between them. However, Islamic historians seem to differentiate between the early khalifate and the later kingdoms, which implies that there is a difference between them. So is there a difference between a Kalipha and a king?
A Caliph can be selected only by Allah but King can be who is not selected by Allah.
Quran in verse 6:57 says:
إِنِ الْحُكْمُ إِلَّا لِلَّهِ
Say, "Indeed, I am on clear evidence from my Lord, and you have denied it. I do not have that for which you are impatient. The command is only for Allah. He relates the truth, and He is the best of deciders."
(The word حکم in this verse (command) has same root and meaning of word حکومت meaning government) and means the command should be issued only from Allah and so only the Caliph selected by Allah can have commands from Allah.
Quran in verse 2:30 says: "I appoint Caliph on earth" إِنِّي جَاعِلٌ فِي الْأَرْضِ خَلِيفَةً
And [mention, O Muhammad], when your Lord said to the angels, "Indeed, I make upon the earth a successive authority." They said, "Will You place upon it one who causes corruption therein and sheds blood, while we declare Your praise and sanctify You?" Allah said, "Indeed, I know that which you do not know."
This verse uses exactly the Arabic word for Caliph and clearly says Allah selects and places Caliph on earth.
Also this verse:
يَا دَاوُودُ إِنَّا جَعَلْنَاكَ خَلِيفَةً فِي الْأَرْضِ فَاحْكُم بَيْنَ النَّاسِ بِالْحَقِّ وَلَا تَتَّبِعِ الْهَوَىٰ فَيُضِلَّكَ عَن سَبِيلِ اللَّهِ إِنَّ الَّذِينَ يَضِلُّونَ عَن سَبِيلِ اللَّهِ لَهُمْ عَذَابٌ شَدِيدٌ بِمَا نَسُوا يَوْمَ الْحِسَابِ
[We said], "O David, indeed We have made you Caliph upon the earth, so judge between the people in truth and do not follow [your own] desire, as it will lead you astray from the way of Allah." Indeed, those who go astray from the way of Allah will have a severe punishment for having forgotten the Day of Account. 38:26
This verse also uses exactly the Arabic word for Caliph. And God says: WE made you Caliph.
There is absolutely no single verse in Quran saying humans have the right to select Caliph and according to Quran only Allah has this right.