As stated in the title, what are the differences between the two? Are they supposed to be entirely different or is simply the next stage of the previous?
Super Saiyan God Super Saiyan, or simply Super Saiyan God SS is the result of a Saiyan gaining the power of Super Saiyan God and then transforming into a Super Saiyan.
This form is physically identical to the first Super Saiyan God form, overall body structure being thinner and slightly taller. The only difference being that the hair is similar to the Super Saiyan transformation but in a blue color. The aura; as opposed to the red-orange aura the previous God form had, Super Saiyan God Super Saiyan boasts a vibrant, flame-like blue aura. Additionally, electricity discharges around the user, signifying the power increase.
So SSGSS is the next stage of SSG.
Besides the physical aspects and the power increase already mentioned, super saiyan blue puts more stress in the saiyan body, another difference which was shown in the series but not in the movie, it's the super saiyan god (red) has regeneration habilities, when wounded by Beerus who introduced half his hand in Goku's body and knocked him almost unconscious, he heals the wound completely after a few minutes. These features if made intentional and not casual are very interesting because it gives room for all kinds of new techniques in the future, one of them is already shown in the last chapter of the manga by Vegeta.