“Will” is a more casual way to make the request.
Will you close the door, please?
“Would” is a much more polite way to make the request.
Would you close the door, please?
You need me to lend you my car keys.
Will you lend me your car keys?
You think your partner should listen to you more.
I would like you to listen to me more.
You need someone to turn off the lights.
Will you please turn off the lights?
You want your best friend to tell you the truth.
I would like you to tell me the truth.