something about the relationship between participle used as adverbial and ellipsis in adverbial clause

I feel that there is a close relationship between them and I wonder whether they are actually the same. When you say "When the mouse saw the cat, it ran off", you can also put it as "When seeing the cat, the mouse ran off" and "Seeing the cat, the mouse ran off". So, you just omit the conjunction as well and you get a participle used as adverbial.

This confuse me because instinctively I think they should be distinctive grammar constructions, but they are so similar, the different is just one word.
So how to explain this?

Thank you in advance!