Let me do it. This is a set phrase that typically precedes a polite action or gesture. "Allow me," my date said before opening my car door. Allow me—I'll open the wine bottle.
allow meand permit me
Please let me help you. (*Typically said by someone politely assisting another person, as by opening a door or providing some personal service. In Allow me, the stress is usually on me. In Permit me, the stress is usually on -mit.) Tom and Jane approached the door. "Allow me," said Tom, grabbing the doorknob. "Permit me," said Fred, pulling out a gold-plated lighter and lighting Jane's cigarette.