let: may imply a positive giving of permission but more often implies failure to prevent either through inadvertence and negligence or through lack of power or effective authority: the goalie ~ the ball get by her.
allow: implies little more than a forbearing to prohibit or to exert this power: a teacher who ~s her pupils to do as they like.
permit: implies express willingness or acquiescence: the park ~s powerboats on the lake.
suffer: can be close to allow or may distinctively imply indifference or reluctance: ~ed themselves to be photographed. indifference无所谓
leave: stresses the implication of noninterference in letting or allowing or permitting and can suggest the departure of the person who might interfere: ~ them to resolve the matter among themselves.