I think while we all feel "different" sometimes, and it can be in a negative way, or in a positive way. I don't know if every type feels like a weirdo as part of their self-concept though.
Everyone takes their sense of difference, well, differently. Some NT's feel different when they see how much smarter they are than everyone (Not all NT's are smarter of course). Usually they are not guilty about this, they take pride in it. Not just that though, sometimes their objective view of the world can make it hard for them to fit in and relate to others, and this can make them lonely. NT's need love too! We NF's sometimes like to think we're special, and we place a lot of emphasis on individual difference, and we like this usually, because we find difference gives the world humanity. However, we are also self-conscious about being different and it is very important for us to have connection to people, so if we don't fit in it can be very upsetting for us. Some NF's will turn this tragedy int a point of pride and equate their tragic flaw as something that makes them special, some will just feel like crap. I have no idea about SP's, but they seem to be the cool ones in-step with things, but they must feel similar things to this as well. I am sure SJ's have felt out of place, but they usually have such a strong grasp on how to be in place that it might not come up as much. When it does I am sure it's awful for them, since so much of their identity is staked on normalcy. But even if once in a while SJ's felt weird, I don't think that's how most SJ's would characterize themselves.