I'm eyeballing Stellaris as a "replacement" game to my Sins of a Solar Empire. Honestly I'm looking for a space "trading" game with only PvE gameplay. Not sure if that even exists.
Not sure I understand the term space trading game. If you literally mean trading, where you haul freights and buy low/sell high in a living economy, which is also PvE, I recommend X3 or X4 (which was just released). X series also involve in empire-building, where you build your own stations and compete in the universe. At its core, it is a space sim, not strategy.
How does Stellaris compare up to Sins of a Solar Empire (SoaSE)? What makes it different [better?] from SoaSE? Is there any other game out there like this? I've played Galactic Civilization III as well.
Sins of a Solar Empire is an RTS game. It also involves a lot of micromanagement of controlling individual ships and directing it to attack etc, while building up your economy and buildings as you would in any other RTS games.
Stellaris is a different genre and there's no "better" or "worse". It is however different in the sense that you are playing it at a greater sense of scale than Sins. Map can contain as much as 1,000 star systems. A single game can also last twice or three times longer than a usual round of Sins. There are no objectives, but there are victory goals which you can pursue as an option. That does not end the game however. It is ultimately a sandbox.
Greater scale as in no direct combat, and you are managing an empire, where Sins feel like you are managing a faction. Sins is also much faster paced because of its own smaller scale.
Colony and economy management is not available in Sins, because of its own scale as a space strategy combat / territory building concepts. Your sole goal is supremacy in Sins which almost always is gained through combat. In Stellaris, that doesn't have to be. You don't have to aggressively expand as that's not your sole purpose. You can decide it to be, though. You also delve into what the game calls "ethics" and that plays a huge role in what your empire's characteristic is to be, and how your people and your colonies operate.
You can also create and customize your own empire. Name it whatever you want. There are many mods out there to further expand your customization horizons.
There are so many more but others would point them out.