In general, lower interest rates will allow you to borrow more money and still pay the same monthly payments that you did with the higher rate.
Beware, though, that interest rates may fluctuate while you are shopping for a loan, so you need to know when the lender you want to get a mortgage from can lock in your interest rate. This lock in from the lender guarantees that you get a specific rate. Rates are usually locked for a specific period of time. If your closing is delayed, you may need to negotiate with the lender for a new interest rate.
Lenders look at your credit history when offering interest rates. It is no surprise that the borrower who pays a 35% down payment and has an unblemished credit history will get a lower interest rate than a person with a tiny down payment and credit problems.