I've been quite a bit intrigued at times when certain persons are perceived as insensitive or even apathetic, even when they seem to be doing reasonably well helping the people or causes they're accused of being apathetic to.
Listening to the book 'Humankind' (an excellent ode to positivity, which takes apart all the negative 'tenets' we go by) by Rutger Bregman, the portion in the Epilogue dealing with compassion, clarified the issue a lot.
The author says that, to really be able to help others, people need to be more compassionate, not necessarily (or not just) more empathetic. This is supported with evidence from a scientific study of people's brains being scanned while they were told to consciously experience empathy and then compassion about a certain situation - it was reported that different areas of the brain 'lit up' in the two cases.
The author gives the example of a child who's afraid of the dark - to help the child, the parent or caregiver does not cower in a corner thinking about the causes of the fear (empathy), but instead feels for the child and soothes him/her with comforting behaviour (compassion).
It seems to me that people displaying compassionate behaviour have to retain a level of reasoning in their mind, to be able to formulate responses and actions to help out, and that may be why they're perceived as 'dispassionate' and thus 'devoid of feeling', not being able to balance 'thinking' and 'feeling' to the degree possible for the more voluble (who may be perceived as more empathetic but may or may not be more compassionate).
Added to this is the factor that compassionate behaviour may be more 'action-oriented', while empathetic behaviour can be more expressive (verbally or otherwise). And we all know that the old adage of 'Actions speak louder than words' has been turned on its head in the current age of 'in your face' behaviour and microsecond attention spans where, for instance, love has to be expressed in words rather than just conveyed through gestures and actions.
Which is probably why people with genuine compassion, who may actually act on their feelings of empathy by helping through their actions, may be unfairly characterised as unsympathetic.