"Let's go" is just that - a suggestion to leave. "Come on" will often be used in the same context, but is more general. It means to get organised. So 'come on, let's go' is the natural order - get organised, finish what you are doing, and then let's go. I would disagree with Kevin's comment that "Come on" sounds more encouraging; "Let's go" sounds more snappy and impatient. Let's go is often just a suggestion. "Come on" on the other hand usually means that the speaker is getting frustrated with waiting around.

Come is used when you order or ask some one to tome to you, you say: come to me, or come on here, or come to my office but Let's go is user when you offer or ask someone to go ( with you ) somewhere. it means that you together will go somewhere.

They both have the same two usages: proposing that the group move to the next location, and telling the group to start working or to work harder. There is a difference in nuance between the two for the second usage (telling people to work harder) though. "Come on" sounds more encouraging; "Let's go" sounds more snappy and impatient.