[kjam22]

Christianity believes that Jesus Christ is the God of Abraham. Islam doesn't. This is the fundamental difference.

If Islam claims to worship the God of Abraham, and Isaac... why did they keep the crescent moon as their "logo"?

[Wm Bach]

why did they keep the crescent moon as their "logo"?

Same reason the jews keep the Star of David around... ancestor's worhip of Ishtar.

[Non-Sequitur]

Christians belive that Jesus Christ is the son of the God of Abraham. If Muslims refuse to believe it then I would point out that neither do Jews.

The crescent moon is symbolic of the fact that it is the appearance of the crescent moon that marks the beginning of the Muslim month. If that means that the Muslims worship the moon because their calendar depends on it, do Christians worship the moon because the date for Easter depends on it (Easter falls on the first Sunday after the first full moon after the vernal equinox).