Bi and pan are very similar terms. Some non-monosexual people prefer one term over the other for various personal reasons. Some people may also use both terms interchangeably to describe their sexual orientation.
The main difference between bisexuality and pansexuality is simply semantics.
Bisexual people usually describe themselves as being attracted to both like and different genders. Bisexual people may find themselves attracted to different qualities in people of different genders. Some bisexual people may find that they are mostly attracted to femininity in women and masculinity in men. Others may only be attracted to feminine women, and more feminine men. Perhaps some are attracted to men emotionally, and women physically. There are many, many different ways to be bisexual. Some non-monosexual people also identify as bisexual because it is a more recognizable term than pansexual.
Pansexual people describe themselves as being attracted to people regardless of gender. These people tend to think of their attractions to people as having nothing to do with the person's gender. They can be attracted to people regardless of their gender expression.
You should identify as whichever you feel best describes your sexuality. It is okay if you use both interchangeably, or just one term. It is also okay if you start using one term, but then later realize that the other fits you better.
Regarding your other question, if your family seems supportive of the LGBT+ community, it is probably safe to come out to them. If you live in a more liberal city, it is also probably safer to come out than if you live in a more conservative city where people still have backward beliefs about LGBT+ people.
Is there a local LGBT+ organization at your school or in your community? If so, you could contact them to meet other LGBT+ people who can help to support you as you decide to come out to other people in your life. It is always a good idea to have a support system in place in case things go wrong when you come out to your family.