Will I be treated differently from funded students if I attend graduate school without full funding?
Confession:  I've tried this myself TWICE in the field of mathematics, so what I say comes from my own experience.

You need not worry about the existence of a caste-system among graduate students.  You will not be treated any differently than any other student if you are accepted into a program and not funded.  The main question is whether or not you will get into the program in the first place if you have no external funding sources...

In science-related graduate schools, it is quite often the case that students will not be accepted into the program unless they have some sort of support (i.e. department assistantship, scholarship/fellowship, etc...).  Students who try to do it all on their own often find themselves under even more pressure than a funded student.  On top of trying to pass extremely difficult courses and pursue original, cutting edge research, they may find themselves also working multiple unrelated jobs that barely make ends meet for rent, much less tuition and all other debts incurred along the way.  Often, unfunded students succumbs to financial pressures and drop out to pursue more financially stable opportunities.  

Students dropping out of graduate programs also make their host departments' statistics look bad in the eyes of their superiors (i.e. deans, university president, provosts, etc..) and can lead to diminished support for those graduate programs.  Since universities don't want to hurt their own reputations (or lose state/donated funding), they tend to be selective of their graduate students.  And I believe this is a major reason why self-funded students are often not even allowed in graduate programs:  statistically speaking, their success rate is likely too low to merit taking a chance.

Of course there are exceptions (e.g. having wealthy parents, pursuing non-science graduate programs, education doctorate degrees often earned by people who work full time as teachers), but it is certainly a red flag if a student willingly tries to pursue a graduate degree in the sciences without any source of funding.

My advice to you:  If you're offered funding, take it!.  If you are accepted into a graduate program and are not offered funding  and don't have any other source of funding apart from yourself, then don't try to do it all on your own.  The sheer cost of graduate school, combined with the uncertainty of you graduating from the program, along with the nightmare of trying to pay off student loan debt for the rest of your life (even bankruptcy will not save you from student loan debt); it's just not worth it to you.