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.
My path through math graduate school has been as follows: 1 year unfunded in Ph.D. program at school A, 1 year funded in Ph.D. program at school A, move to different school and spend 4 years funded in the Ph.D. program at school B. So I've spent time in Ph.D. programs both as a fully funded "regular" math grad student, and as an unfunded math grad student.
Being an unfunded grad student had a couple of potentially negative effects (I won't discuss the obvious financial burden):
I felt a bit disconnected from the math department. The funded grad students had shared offices, which naturally led to them getting to know each other. As an unfunded student, I had no office or access to any shared department space (break room/kitchen area, etc.)
My goals were slightly skewed from what they should have been. I went in to my first year as an unfunded student with this feeling that I needed to do "better" than "everyone" else. The reason for this was that I wanted to secure funding for the subsequent years. I basically met this goal, but in retrospect, doing better than my fellow students in all of my courses was not the best goal to have for a first year of graduate school.
To help remedy pont 1, I worked as a paper grader in the math department my first year, while unfunded. This helped me to get to know a few professors better, which I'm sure didn't hurt when I applied for funding the second year.
As for being treated differently: I don't think I was ever treated badly or differently just because I was not funded. There is a range of different types of funding amongst students: Departmental TA, faculty-funded research assistantship, department fellowship (with no teaching, say), NSF fellowship (in the U.S.), unfunded, partial TA (tuition waver, fewer duties, lower pay) etc. Some or all of these might exist in any given department, and some may be considered "more prestigious" than others, but in my experience these differences don't lead to a class system among the graduate students.
I tried this. Do this only if you do not have to get a job to support yourself during school. If you do have to get a job, you will probably only burn yourself out and fail. If you do not need to work to support yourself (you're rich; your parents are rich; whatever), go ahead.
Of course, a T.A. or an R.A. is a type of job, but it's a lot different than a real job.