Can someone explain to me, the difference between "Seeing" and "On seeing"?
Is "Seeing" a gerund?
On seeing him, he moved faster. Seeing him, he moved faster.
Are these two sentences both grammatically and semantically perfect?
There is a lot of ambiguity between "gerund" and "present participle". At least some grammarians think that the terms are out-dated. There are just "-ing words", which might function a little like a noun in some contexts, and a little like an adjective in another.
However, you can parse "[On] seeing him" as a gerund. It is filling the place of a noun. For example, you could say "On Tuesday". In the other sentence "Seeing him" is a participle phrase, since you could not use a noun or a noun phrase in that position.
But a participle phrase or a prepositional phrase can both act as a modifier. And these two modifiers have a similar meaning. (If I had to find a difference, "on seeing him means "at the time of seeing him" and "seeing him" means "as a result of seeing him", but the sense is so close I can't really separate it)