After seeing him crying, ... . The participle in bold is an objective complement; it completes the meaning of the object (him) of the verb seeing.
After seeing him (to) cry, ... . The infinitive clause, whose (to) is dropped, functions as an objective complement.
Both complements are non-finite clauses, which means they are not inflected for tense, person, and number.
What is the difference between the two forms? The crying form means you saw him in the act, part of her act of crying. On the other hand, the infinitive form denotes the whole process of crying was witnessed.
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