The only difference is that Moses faced more doubt from his people then Abraham did.
Although they were similar leaders, they did have a different relationship with God. Abraham from the onset did not question God at all. Anything that God told him to do, it was done. For example, God asked Abraham, "Take your son, your only son, Isaac, whom you love, and go to the region of Moriah. Sacrifice him there as a burnt offering on one of the mountains I will tell you about" (Genesis 22:2). Abraham did not question why God wanted him to sacrifice his one and only son; nevertheless, he did not question God and preceded with him command. When they got on the top of the mountain, God saw that Abraham truly had faith in him, thus he sent a ram instead so that Abraham did not have to sacrifice his son. Abraham never had a doubt in God's will and was offering even to sacrifice his own flesh and blood to prove his undoubting faith. Moses on the other hand, questioned God. When he first encounters God in the form of a fiery bush, Moses questions God's will by asking, "Who am I, that I should go to Pharaoh and bring the Israelites out of Egypt?" (Exodus 3:11). Although he meets God in a physical form, he still questioned him with doubts of his own abilities. Unlike Abraham who had unquestioning faith, Moses started out his relationship with God with a lack of faith. Eventually, Moses does develop the same undoubting faith that Abraham processed. Only initially did he have doubts about his relationship with God.
Finally, they both received similar covenants, however they communicated with God in a different way. God told Abraham, "Leave your country, your people and your father's household and go to the land I will show you. I will make you into a great nation and I will bless you; I will make your name great, and you will be a blessing. I will bless those who bless you, and whoever curses you I will curse; and all people on earth will be blessed through you" (Genesis 12:1-3).