My university has a 5 year BS/MS degree for mathematical sciences and I am graduating a year early, so I can do the BS/MS in only 4 years. I am considering applying to PhD programs at other universities after I complete my MS. Do PhD programs want students right out of undergrad? Or will I be at an advantage showing that I can take grad courses (assuming I do well)
My answer reflects the situation in the US and Canada. In the US, most Ph.D. programs accept people directly from undergrad; at a few US and most Canadian schools you'll be expected to start in a master's program, but it is generally relatively easy to move up to the Ph.D. program (not significantly harder than avoiding getting kicked out of most US Ph.D. programs). Whether it says "MS" on your degree will matter less than what courses you have taken. Having another year of coursework will probably strengthen your application (or at least gives you a chance to), but if you have to pay a significant amount of tuition, then I have serious concerns about whether it will be worth it. If you can get into a funded master's or Ph.D. program, then that will save you a LOT of money, and you can get roughly the same boost to your future applications spending a year there that you would by being in the BS/MS program.
EDIT: Since the OP has clarified that the extra year would be funded with a stipend, my comments above aren't really applicable. In that case, it probably is the best plan to stay at the current university and get the BS/MS.
Maybe I'll just make a general comment here which seems relevant: its generally pretty advantageous to stay longer at the current stage of career where you already are, as opposed to moving on in a rush. Our culture makes a powerful mystique around completing things quickly, but all other things being equal, it is to your benefit to be a little ahead in actual accomplishments of where you are in your career (which extra time is one way of accomplishing). You get "clock resets" at various points of your career (when starting or finishing a Ph.D., for example), if you can get more done before those, all the better. Of course, often financial or personal considerations have to be weighed against this.
Some of my friends went to a U.S. university that had many students do BA/MA and BS/MS programs ("coterms"), and for the most part it seemed like a good chance to spend a little more time to do more graduate-level coursework. I hadn't asked what their funding looks like, but (especially if you'd be graduating in 4 years) there might be a way to make any financial aid or scholarships help cover the coterm.
I'll emphasize two potentially related benefits to this:
If you have classmates and roommates you like, this is a way to spend a "senior year" with them, which could be personally very meaningful.
Let's say you got into an equivalent 1 year MA program somewhere else. It would require you to relocate, create a new support structure, learn new systems, change your email address, etc. You will have to do that sometime, but for many people that would be important to take into account if you plan to be there for only one year and then move on.
Those "touchy-feely" considerations mean you may be more likely to succeed at your graduate coursework (e.g. through fewer life disruptions, unless everyone around you starts partying heavily). You can also continue to build relationships with professors, which will help you in applying to schools and in calling on them as mentors over time.
TL;DR: If this school is working well for you for your BS, then you have strong priors that you can succeed there for your MS.