Nobody really has the power to give a vastly better rate. Mortgages are bought and sold on a secondary market in large bundles and as such they are all about the same. The main differences you will see are the costs associated with originating the loan. A typical origination fee is .5-1% and you can expect to see some sort of processing fee $250-$1000.
Points are 1% of the whole mortgage 1 point on a $250K mortgage is $2,500. They are typically used to buy a better rate up front. You pay more at closing and the lender agrees to give you a better rate than the standard rate for your risk profile.
A GFE is an estimation of all of the costs associated with obtaining the loan in question. You will want to get these from all of the lenders you talk to and compare not only the monthly payment but also the fees.
I hope that helps.