Lasciare vs. permettere
While both 'lasciare' and 'permettere' can mean 'to let' or 'to allow' in Italian, they have distinct uses and connotations.
Lasciare
'Lasciare' primarily means to leave, let go, or allow something to happen passively.
Lascia che i bambini giochino in giardino.
(Let the children play in the garden.)
Ho lasciato la finestra aperta.
(I left the window open.)
Lasciami in pace!
(Leave me alone!)
Permettere
'Permettere' implies giving explicit permission or actively allowing something to occur.
Il capo mi ha permesso di lavorare da casa.
(The boss allowed me to work from home.)
Non permetto che si fumi in casa mia.
(I don't allow smoking in my house.)
Mi permetti di farti una domanda?
(Do you allow me to ask you a question?)
Summary
'Lasciare' is often used for passive allowance or leaving something as it is, while 'permettere' is used for active, explicit permission. 'Lasciare' can also mean 'to leave' in a physical sense, which 'permettere' cannot.