The difference between Koro-Sensei and the God of Death (No. 2) and the humans with tentacle cells is that Koro-Sensei's cells fully turned into "Tentacle-Cells".
Kayano and Itona only have implanted Tentacle-Cells in their bodies; they form, as mentioned in the question, another organism in a symbiotic relationship with the human body they are attached to. The pain created from the tentacles is only present in the "human"-part of the organism. The tentacle(-cell)s are something that doesn't naturally belong to a human body and the body wants to get rid of the foreign cells (the same problem exists with transplantations of organs) which causes huge pain. The only way to avoid this pain is through "medical help" (like medicaments for real transplantations).
In Koro-sensei's case, all of his body cells transformed into tentacle-cells, so his body is only made up of one organism. He doesn't have a "human body" anymore in which he would fell the pain of the rejection of the tentacle cells.
The conclusion is: the human body of the God of Death (No. 1) fully transformedinto Koro-sensei's body. In Kayano's and Itona's case, something has been added to their bodies.