I allowed him [to play].
 I let him [play].
The bracketed elements are both infinitival clauses.
The grammar is the same in both examples, as is the meaning. The only difference is that "allow" requires a to-infinitival complement, while "let" requires a bare infinitival complement.
"Allow" and "let" are catenative verbs and the subordinate clauses "to play" / "play" are not objects but catenative complements.
Syntactically, "him" is direct object of "allow" and "let", and the semantic (understood) subject of the infinitival clauses.
"Him" is called a 'raised object' since the verb that "him" relates to syntactically is higher in the constituent structure than the one it relates to semantically.