If I ask a woman to be my wife and she accepts that and I accept her to be my wife and the God is the witness on that. Is that possible and the marriage is not temporary? What is the difference between the marriage in the Islamic way and other ways, such as Christian?

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.

The difference between an Islamic marriage and an un-Islamic marriage is that the Islamic marriage fulfils the conditions and pillars of a correct religious marriage, the most important of which are: 

Therefore, if a man proposes to marry a woman –as you mentioned in your question-, and she agrees and he marries her, then their marriage is void even if he gives her a dowry because it did not fulfill two major conditions: the consent of the guardian and the presence of two witnesses. The Prophet, sallallaahu ‘alayhi wa sallam, said: “There is no marriage except with a guardian and two trustworthy witnesses.” [At-Tirmithi and Abu Daawood]. The Prophet, sallallaahu ‘alayhi wa sallam, also said: "Any woman who marries without the consent of her guardian, her marriage is void, her marriage is void, her marriage is void." [At-Tirmithi and Abu Daawood].

Allaah Knows best.