How much do budgets affect likelihood of grant funding?
A badly written budget could put your proposal out of contention, but a well written budget that scrimps on costs isn't likely to help you get funded (what are the odds that your project is "on the bubble" for funding and just fits within the total funds available?)  

In your proposal you should ask for what you need to get the research project done and to support the "broader impact" of the proposal.  Don't load up the budget with unnecessary expenses just so that you can get a larger award.  On the other hand, don't prepare a low-ball budget that doesn't include adequate funding for travel, students, etc.  Being too extreme in either direction can hurt the reviews of your proposal.  

Be aware of your institution's overhead rate and if it is higher than normal, don't be surprised if your budget is larger than average for the program that you're applying to.  If your institution has a relatively low overhead rate, than don't be surprised if your budget comes out on the low side of average. A lot of the variation in budgets is due to differences in overhead rates (along with PI salaries and the cost of students.)  

Also, work with your "restricted funds" office to make sure that your budget conforms to your institution's regulations as well as the applicable federal regulations.  

In my experience, the likelihood of getting funded seems to have little connection to the size of the budget, as long as the budget is within the normal range for the program, the expenditures make sense for the project, and there aren't any very expensive looking items.  

For a grant in mathematics you really should have a very a simple budget and it shouldn't be hard to justify the budget.  Compare this with some grants I've worked on with earth scientists that included international travel to places like Peru and Tibet, shipping boxes of rocks internationally, expensive laboratory tests, etc.