"I feel like you don't care about me."
What I feel cannot be argued with; if I feel angry, I am angry.
However, what I believe is open to critical analysis. What I believe may be true or false.
When I say, "I feel like you don't care about me," what I'm really saying is, "I believe you don't care about me," and I happen to be feeling negative feelings in response to this thought.
That belief could be wrong, and, if the subject matter of the belief is someone else's internal state or attitude, that belief is very much open to confirmation or rejection by the individual in question.
"I feel like I'm not being heard."
To "hear" means to perceive sounds through the ear.
When people say they're not being heard, they usually do not mean this.
Rather, this usually means, "You have not obeyed me," or "You have disagreed with me."
Clearly, it's reasonable, when in a discussion, to expect the other person to physically listen to what you are saying and cognitively process it.
However, it's not reasonable, when in a discussion, to expect the other person to agree with everything you say and obey all of your commands.
So, next time you want to say, "I feel like I'm not being heard," reword it as: "I have observed that you disagree with me / have not obeyed me." Then see where the conversation goes.
If someone says, "I don't feel like you're hearing me," it's reasonable to say, "I have heard you. I simply disagree with you / have not complied with your request."