As with these two sentences, translating them literally will give you a good idea on their nuances: the first one translates to "let's all go together (everyone)"; the second, "let's go together". The first specifically refers to the entirety of the group/crowd/people in question. "모두" means "all", so you'd use it to address, literally, "everyone", "altogether". Depending on the context, "같이 갑시다" may connote the entirety but not necessary; it may just be used when asking someone to accompany you or let you accompany him/her/them. People may also note the term "모두" in the first sentence imbues that tone/feeling of "togetherness". Hope this helps.
