One difference that comes to mind is whether we use these words to refer to animals or children. If you put them in the following order - rear -- raise -- bring up - the difference is clearer. 'Rear' is mainly used for farm animals. It is used for children, too, but it's less common. 'Raise' can be used for either animals or children. For American English speakers, 'raise' is the verb of choice when referring to children. For example, US citizens will often proudly refer to themselves as being "born and raised in [a particular location]". This is a particularly American-style expression. British English speakers might sometimes use 'raise' in this way, but it isn't very common. 'Bring up' can only be used for humans. British English speakers tend to use this word more than American English speakers do. For example, a British English speaker would be more likely to say "I was brought up by my grandparents" to refer to the people who cared for us in our childhood and "I grew up in Scotland" for the location. Just an observation on my part. Others may disagree.

There is no real difference in meaning when taking about children. 'Raise' and 'rear' are slightly more formal, that is all, with degree of formality arguably rising from 'raise' to 'rear', but we tend to use them all interchangeably. You may see in psychological or scientific papers that 'bring up' is not generally used. However, when it comes to animals, you would never use 'bring up', but 'rear'. I have very seldom heard the use of the word 'raising' when it comes to animals, but it is used occasionally. The farmer rears cows. My daughter raises chickens on her farm. The calf was hand-reared.