Having played Saints Row 1-4, I have to say the series felt like it had an identity crisis moving forward.
Saints Row aged better than the 3D era GTAs. It wasn't a crime drama, it was a rise to power like storyline with funny moments. There are three gangs with different attributes, with the Saints being on the bottom. So, you take over the city by sabotaging rival gang operations and conquering strongholds. The story wasn't too serious or too outrageous, it had a great middle ground that left me entertained and wanting more. For a "GTA ripoff", Saints Row held its own.
Saints Row 2 I felt was darker in tone than SR1. The boss could speak, but he was more of an insensitive asshole. The story and setting I felt took itself too seriously (i.e. Stilwater being "expanded by some high corporation) and the earlier gangs were missing. The new gangs in 2 felt barebones compared to 1. However, SR2 is more accessible than 1. The activities you could do, while memorable, distracted from the bleaker tone of the story line. Septic Tank and Insurance Fraud were my favorite.
Saints Row: The Third. Interesting direction to say the least. I guess they were trying to differentiate themselves from GTA, so they went for a campy, arcadey tone. I can respect that, it does feel more consistent. It was more fluid than GTA IV; the zaniness reminds me of the fun from 3D GTA. I liked the new characters and the Boss being less angry. It was easier than 2 and 3. Having played SR3 before 1 and 2, the tone change was disorienting at first. I didn't hate it compared to other SR fans.
Saints Row IV, imo, is the best SR game yet. It cranked up the obscureness to 11, while being intelligent and thought provoking with its humor and story. It did add a recap of how all the SR games come together in continuity. It brings it all together in one universe. The superpower aspect made it different from other open world games. The humor and self referential content was the highlight of my experience. Enter the Domatrix was short, but the gaming tropes it added was a blast.
I'm interested with how and where the next Saints Row game will go. It could go either way, but one key thing it needs to do is continue to be different from GTA.