Filing by mail requires you to print and assemble all of the forms required to process your income tax return.  If you mail your federal return and are expecting a refund, you can start checking the status of your return on the IRS website Where's My Refund 4 weeks after mailing your return. It takes that amount of time for it to show up in their system, sometimes a bit longer if they are backed up. 

E-filing your return does not require printing and assembling your return.  Last year, 9 out of 10 e-filers received their IRS tax refund in less than 21 days,