Monoids form a non-full subcategory of semigroups
The category of monoids forms a subcategory of the category of semigroups via the forgetful functor that sends a monoid to its underlying semigroup (forgetting the neutral element). However, this subcategory is not full.
In other words, the functor is faithful and injective, but not full.
The functor from monoids to semigroups is injective because the binary operation on a monoid determines its neutral element.
A homomorphism of monoids is completely described by its underlying set map, so it is definitely described by the corresponding homomorphism of semigroups.
Lack of fullness
There can be homomorphisms of semigroups between monoids that do not preserve the neutral element. For instance, we can have a homomorphism from the monoid of integers under multiplication to itself, that sends every integer to zero.