As a direct study of the mechanisms of the brain, neuroscience most definitely is the same as "the science of mind and behavior". Mechanisms of the brain are what makes up the mind. And given that neuroscience is taught in basic psychology courses at universities as one among several "schools" or methods of psychological research, then yes, it is psychology.
I personally would argue that even "softer" schools of psychological research are good science as long as they are able to form testable hypotheses and are scrupulous in how they conduct experiments. Even though human behavior at the individual level is more erratic than, say, gravity, if psychologists are able to determine that humans in a representative sample are 70% likely to respond in a certain way in a certain situation, and these findings can be consistently replicated by other scientists, then that is still useful, scientific knowledge. Even when a whole paradigm of psychological thought is essentially proven wrong, like behaviorism was, the fact that it could be proven wrong testifies to its adherence to scientific principles, and the fact that it was proven wrong is also useful knowledge. This is an important part of scientific progress.
Freudian theory, in contrast, can't be proven wrong. There is literally no way to prove it wrong, as there is no conceivable behavior in any conceivable situation which it can't explain in hindsight. And that's why it's not good science.