How does the PhD experience differ for paying students?
There are usually 4 ways to pay for a PhD in my experience in the US:


Out of pocket with your own money or with loans
Get a fellowship or scholarship
Be a Teaching Assistant
Be a Graduate Research Assistant


None of these are necessarily mutually exclusive. TA and GRAships usually come with a whole or partial tuition waiver or payment of tuition. Fellowships and scholarships may come from the university, another government entity (e.g. the US National Science Foundation or Department of Energy), or a private source. 

In my estimation, a GRA or fully-funded private fellowship is the best way to pay for graduate school for those whose focus is research because there are no teaching or grading responsibilities to take time away from working on your research. If you intend to go on to a professorship, starting with a TA for the first few semesters or years may help you learn something about teaching, but I wouldn't have wanted the TA responsibilities during the time I was writing up my dissertation.

Myself, I had a GRA with my full tuition paid plus a small privately endowed fellowship administered by my university which supplemented my income. Also, at the time, having a GRA position gave me health insurance. TA and GRA jobs are usually limited to 20 hours a week during the long semesters in the US since you are likely to be in classes at the same time. My advisor frequently upped me to 40 hours per week during the summer and winter breaks to supplement my income further. 

Paying your own way, whether with fellowships, loans, or your own personal savings means that you aren't being paid by the advisor. A GRA position may be required to focus on a particular research project that funds the existence of the position. If the money comes from their start-up package or a private source, then you may be given much more freedom in your choice of research work, but either way, the GRA position is a job that allows the advisor to direct your work much more than if you fund yourself.