It's probably best to start out with bullying given that bullying is so widespread in our society. Bullying happens in school. Bullying even happens in kindergarten. Bullying happens in the workplace. It happens in the community. It happens in politics.
It would be simple to state, if you don't understand the issue, that capitalism as a system is what legitimizes bullying. But see capitalism is only part of the issue here and what makes capitalism the issue is the confusion many people have over money. Many people do not seem to understand that money, like time is a measurement of value simply because we do not spend money on things we do not value. For many people, far too many people, money is a symbol of power and wealth and this is where we get into the issues of bullying, exploitation, and the struggle for social, economic and political power.
But see when you're out walking among trees, say in a park or a forest, you don't go round judging trees, do you? You don't go round calling trees fat, or stupid, or ugly or do you? You don't go round telling trees to fuck off or leave you alone, or do you? You simply accept the trees for what they are and appreciate being around them.
But see when you get human beings around other human beings most people start becoming more judgmental.
Bullying is where perpetrator and victim know each other
Usually in an instance of bullying there's an existing social relationship between the perpetrator (bully) and victim. This is also the case in many cases of abuse where there is a relationship between the abuser and the victim of the abuse.
This is generally not the case when it comes to social stigma because all too often the people involved do not know each other. This is the main issue with hate crime legislation because in the case of social stigma there is often no pre-existing social connection or relationship between the perpetrator and victim and in the absence of such a relationship it can be very difficult to prove hate to define a violent crime or an attack as being motivated by hate.
You see prejudice and discrimination which is motivated by social stigma (and not hate) can be expressed in subtle and not so subtle ways.
Social stigma can be expressed openly towards someone who fits a label through mocking, ridicule, verbal attacks, and even physical attacks. Unlike bullying where the victim can identify the bully or perpetrator if the hostility is motivated by social stigma then the hostility or attacks can come from anyone and it may not always be clear or apparent why.
Often the hostility is not so active or aggressive and manifests only as social disapproval or scorn, or labelling. But often while the hostility doesn't result in a verbal or physical attack the social disapproval can come from more people. Some people might put up barriers or make things more difficult for you on the basis of their social disapproval of who you are.
Then you have the really passive or subtle forms of stigma and discrimination, such as avoidance, shunning, aversion, which is often much harder to make out, often because this comes with excuses.
The main difference between bullying and social stigma is whether you can deal with the issue or not. In the case of bullying while there are different approaches you know the bully and know exactly who you're dealing with. This is generally not true with social stigma, when you have negative social encounters with different people who you don't know. If you're affected by social stigma you might not have the awareness, the energy or the resources to even try to attempt to resolve the issue.
You're effectively disabled by other people's beliefs, prejudices and assumptions.