Answer
In the case you are referring to, both groups chant Hare Krishna. The difference is not in chanting itself but in the exercise of free will.
At every stage of spiritual advancement there is a test. Even senior devotees who occupy important positions can fail that test if subtle material desires remain.
The fundamental issue is the desire to become a guru rather than remaining a servant. This is contrary to the principles of Vaishnavism. In devotional life, one should act only according to the order of the spiritual master and should not independently cultivate personal ambitions—even if those ambitions appear to be for serving the Lord.
This is why the authority system exists. Even if an authority occasionally gives imperfect advice regarding service, the greater principle is to remain within the authority system rather than acting independently.
Real spiritual advancement means giving up one's own desires and accepting the desires of the Lord as they come through the proper authority according to one's capacity.
Strict obedience to the guru's order becomes difficult when subtle material desires remain.
For a devotee, the final snare of Maya is the desire to be worshipped.
For impersonal philosophers, the final illusion is thinking, "I am God." Devotees have already accepted that they are eternal servants of God. Their final temptation is accepting worship meant only for Krishna.
A spiritual master accepts worship only because he has been authorized by Krishna and the disciplic succession to represent the Lord. He does not keep that worship for himself. Like a transparent medium, he passes everything to Krishna, knowing that every soul belongs to Him.
The disciple cannot directly perceive Krishna, so the scriptures instruct him to worship the bona fide spiritual master as the representative of God. Through that worship, the disciple becomes connected to Krishna.
Therefore, accepting the position of being worshipped without proper authorization is contrary to the constitutional position of the soul.
A person may still render devotional service and help many disciples progress, but this subtle impurity exists at a higher spiritual level. It is not an ordinary karmic fault.
Krishna reciprocates according to each person's surrender. Those who follow Srila Prabhupada's instructions in every detail receive a corresponding level of empowerment.
There are many Krishna conscious people in the other camp. Their chanting is effective, and their preaching also benefits others. However, they have missed an important instruction regarding the continuation of Srila Prabhupada's initiation system.
As Krishna says, "As one surrenders unto Me, I reciprocate accordingly."
Therefore, they receive reciprocation according to their level of surrender, but the special empowerment required to continue Srila Prabhupada's global mission flows through strict adherence to his instructions.
The difference is not whether devotees in the other camp are Krishna conscious—they certainly are—but whether they have fully aligned themselves with Srila Prabhupada's vision for maintaining a united worldwide movement.
Likewise, if a temple follows Srila Prabhupada's system for offering food to Krishna, its prasadam is genuine prasadam. We should not deny that.