Bishops as vassals; thats the real rub in the Investiture Conflict. The Ottonian Kings/Emperors (919-1024) decided their only counterweight against the great nobility was the Church.
First of all, a Bishopric was not hereditary, so any lands/wealth they accumulated would not contribute to creating a powerful dynasty that could challenge the royal family for the throne. This also let the King install a loyal candidate of his own choice every time the office became vacant, whereas with the nobility, their heirs expected to inherit their fiefs and revolted if the King interfered.
Second, the Church tended to view the Kingdom & Empire as a whole, ie, they promoted unity, whereas the nobility were loyal to the tribe (Franconians, Swabians, Saxons, Bavarians) and/or their own interests. The clergy was also educated, and so made much better administrators and officials than some guy reared and trained to do nothing but make war. There were a host of other reasons which I wont bore you with, but suffice it to say that the Crown enfieffed the Church on a large scale in order to offset the great Dukes.
In the Emperor's eyes, a Bishop who held Imperial lands was definitely a vassal because those lands belonged to him. But to the Pope, the Bishops were HIS men, and should only be invested by him. You see the problem here.
So all 6 of the German Archbishops were vassals of the Crown, as well as a host of Bishops, and they all owed the Emperor military service. Any Bishop referred to as a "Prince" held land from the King and so in his eyes was a vassal; and thats why he refused to give up the right of investiture. Henry V stated the obvious; as long as those guys hold MY land, they are MY vassals. If they give me all that land back, I will release them from vassalhood. The Pope agreed in theory; but the problem was that the Bishops didnt because they LIKED being princes instead of preachers; they liked being rich, they liked leading armies, they liked being politically powerful...so a great many German, Italian, & Burgundian prelates sided with the Emperor against the Pope and urged him not to sell them out...
So for example, Odo de Bayeux was not a vassal because he was Bishop of Bayeux, but because he held fiefs from the King in addition to being Bishop! So as Bishop he was subject to the Pope; but as Earl of Kent, he was a vassal of the King. But generally speaking, this phenomenon was never as widespread in England and France as it was in the Empire (Germany, Italy, & Burgundy).