Quite simply, to be disadvantaged is to have less advantage than others. For example if you live in a remote area you might be disadvantaged in looking for work or accessing training courses.
To be vulnerable means that you are at risk of something. For example if you don’t have enough food to eat you are more likely to become sick than someone who does have enough food. Therefore you are vulnerable to ill health. Vulnerable and disadvantage are often used to describe the same thing.