I scrolled down to find a thread titled "Fellowship versus RA
" that addresses these questions.
There are two entities involved in a typical physics graduate student's funding: the department, and the research adviser.
The department has a pool of money that it acquires from various sources (the government/state taxes/private donations/endowments/grants etc) and has an annual budget for running its business. This budget includes paying graduate student TAs. When (if) you eventually stop TAing and taking classes and join a research lab, your funding no longer comes from the department, but from your adviser's grant money. Professors typically begin setting up their research labs at a university with the help of a financial offer from the department, but eventually they are like an independent business. They apply for and receive grants from various public and private sources to fund their research and their graduate student researchers. If they can't get enough money, they have to cut back on projects or students.
When you enter graduate school without a fellowship, you'll usually be TAing. This is very good for the department and the professors. The department needs TAs to run their classes and they get a fresh pool with each year of incoming students. You're usually very busy taking classes in your first year or two and so have limited use as a researcher anyway. So, usually you start on department money by being paid to TA while you take classes and start talking to different professors and looking for a research group to join. Once you join, your professor doesn't want you wasting your time with classes and TAing, they want you in the lab. They pay your stipend and tuition out of their grant money. If they can't afford to do that, sometimes you'll TA for your stipend while you do research "for free." This is more common in theory labs where grant money is tight.
If you come in with an external fellowship, first, you are not required to TA for your money (unless it is a requirement for graduation). This frees up your time to either focus on your classes and get them out of the way sooner, or start doing research earlier and hopefully graduate earlier. More than that, you will probably be able to get into any research group you want because you are basically free labor to professor. They get another graduate student without having to use up tens of thousands of dollars a year to pay your tuition and stipend. Without a fellowship, you have to worry about finding a professor that has enough money for another student and then convince them that they should pay to have you above any other student that might also be looking to join that year. With a fellowship, you have the pick of the litter. Plus, it usually pays a bit more than the department stipend.