Your question does not make it quite clear what, specifically, the external and internal motivations are for, so I will discuss the factors which drive their relationship and their desire for independence.
As far as their relationship goes, it is obvious that Lennie and George share a close bond. On an internal level, they enjoy having company and being able to communicate. It is good to know that someone else 'has your back.' The two men are attached to each other because they share a common interest and take mutual delight in discussing their plans. Furthermore, they actually like each other. Each cares equally about the other. Even though George is at times quite harsh and Lennie threatens to leave him, it is obvious that neither really wishes to break company with the other.
External factors which affect the relationship and lead to its creation are the fact that George promised Lennie's aunt Clara that he would take care of him. He is, therefore, bound to his pledge. George sometimes expresses resentment about having to bear this burden, as in the following excerpt from chapter one:
God a'mighty, if I was alone I could live so easy. I could go get a job an' work, an' no trouble. No mess at all, and when the end of the month come I could take my fifty bucks and go into town and get whatever I want. Why, I could stay in a cat house all night. I could eat any place I want, hotel or any place, and order any damn thing I could think of. An' I could do all that every damn month. Get a gallon of whisky, or set in a pool room and play cards or shoot pool..."
"An' whatta I got," George went on furiously. "I got you! You can't keep a job and you lose me ever' job I get. Jus' keep me shovin' all over the country all the time. An' that ain't the worst. You get in trouble. You do bad things and I got to get you out." His voice rose nearly to a shout. "You crazy son-of-a-bitch. You keep me in hot water all the time."
George is clearly frustrated, but does not truly mean what he says, though, and later apologizes to Lennie when the latter threatens to leave.
"I been mean, ain't I?"
"If you don' want me I can go off in the hills an' find a cave. I can go away any time."
"No- look! I was jus' foolin', Lennie. 'Cause I want you to stay with me..."
A second external factor that motivates the two to maintain their relationship is the fact that their closeness makes them different than all the other ranch hands who don't have the privilege of close friendships. The other men are lonely and drift from place to place, never forming any close ties. To George and Lennie, their association makes them special, as is evident from the following passage:
George went on. "With us it ain't like that. We got a future. We got somebody to talk to that gives a damn about us. We don't have to sit in no bar room blowin' in our jack jus' because we got no place else to go. If them other guys gets in jail they can rot for all anybody gives a damn. But not us."
Lennie broke in. "But not us! An' why? Because... because I got you to look after me, and you got me to look after you, and that's why."
The two men share a dream to live independently. They both have a deep desire to obtain land and be masters of their own destiny. Internally, this is probably a universal motivation; every person wishes to be in charge of his or her own life, free of another's control or command. The dream is so powerful that it empowers the two men and deepens their attachment.
Lennie expresses his desire in simple terms: he wants to live "off the fatta the lan'" and tend rabbits, while for George it means that he can decide to do whatever and whenever he wishes. He can work when he wants to and entertain guests and run his life just the way it pleases him. The ideal is a powerful one, and both have a clear visual image of their shared future.
External factors which drive the dream are the drudgery of their existence; they want to be free and in charge, instead of having to rely on others for their existence. Having their own place would mean they can take care of themselves. Secondly, they would be landowners, something that other men in their situation would never have. They would be respected and would be able to make choices about who they did and did not want for company, a choice they do not currently have as ranch hands.
It is tragic, though, that their ideal never comes to fruition, for fate intervened and, in the end, their hope was destroyed.
Lennie's internal motivation was to have a place where he could be free. We see this is his desire to "live off the fatta' the lan.'" And external desire that is easy to see is his desire to touch soft things. He does this with the woman's dress (prior to the story's beginning), dead mouse, the puppy, Curley's wife's hair, and he hopes for this with the rabbits at their dream ranch.
George's motivation is to make it through today and tomorrow defending Lennie. Although he lets Lennie believe that George would be so much better off without him, I think George truly enjoys Lennie's gentle heart and character. He also takes his word seriously and since he promised Lennie's Aunt that he would take care of Lennie, George wants to follow through. These are internal motivations. On the external, George makes it appear as if he would like to have a girl and a good time. Although he probably really desires it all, he has made the sacrifice to keep Lennie around which prevents some motivations from becoming realities.
George's final motivation to kill Lennie in the book has more to do with protecting Lennie from pain than anything else. This is perhaps George's most admirable motivation because even though it seems wrong, it demonstrates responsibility and care.