Preapproval is a much more involved process. The lender will take all pertinent information regarding your finances and perform an extensive check on your current financial status. This will ultimately give you the exact amount that you will be eligible for (depending on what type of loan you decide to go with). Being preapproved lets the seller know that you have gone through an extensive financial background check and there should be no unexpected obstacles to buying the home. You can see how being preapproved would be more attractive to a seller than just being prequalified.
The type of mortgage you apply for will depend on many factors, but the majority of that decision will be based on your ability to pay a monthly installment. If you can only afford a $1000 dollar a month payment, you are not going to go out and buy a $250,000 home, unless you have a large sum of money set aside to make a sizable down payment! Financial planners say that you shouldn't pay more than 28% of your gross income for housing (that includes principal, interest, taxes, and insurance). Depending on your debt to income ratio, that percentage may change.
