How has the Hadith been influential; significance of Hadith?How has the Hadith been influential throughout the history, in Islam and in the secular world?
Anyone who has read the Hadith would surely derive benefit from some or even all of the sayings, though Westerners and Western Christians mostly have not done so. For those who have read it, the Hadith has been influential in the history of the world and in secular circles. There are some sayings that teach what seem to be universal concepts of moral good and therefore coincide with Christian teachings. Two examples are:
God is more compassionate on His creatures, than this woman on her own child.
The best of alms is that which the right hand giveth, and the left hand knoweth not of.
One saying that has been overlooked in Western history as being superseded by primogeniture and the greed that accompanies it is:
A man's first duty should be to his own family, if poor.
... [alms] given to kindred hath two rewards; one, the reward of alms, the other the reward of helping relations.
Here is one Hadith saying that the contemporary world deeply needs to be influenced by:
Humility and courtesy are acts of piety.
Verily God instructs me to be humble ... and that no one should oppress another.
I don't think that you could really say that hadith has been significant either in the secular world or in the non-Muslim world. Since people who are not Muslim do not think Muhammad was anyone particularly special (outside of his historical significance) there is no reason for them to be affected by his sayings. Within the Muslim world, hadith has been extremely important as a source of authority and a way to understand the scriptures.