What type of rails does your saw have? That will definitely help determine which is the best way to go. In the end it is all personal preference and opinion.
My old Ryobi table saw had a separate router plate wing, and it worked OK, I liked being able to use the same fence for both machines. I definitely had to plan my work so I wasn't going back and forth with my set ups all the time.
My new table saw actually has MORE surface area than the Ryobi, and I find that I have gotten into the bad habit of using that extra real estate as a staging area for the different pieces of the project, or as an assembly area. I definitely have room for a router in there, but I really like having the extra work surface.
My router table is not typical, it is basically an assembly table/tool cart/storage unit that I can roll all over the place. I can even mount my daughter's mini lathe on it, so it serves multiple purposes.
My favorite reason for having a stand alone router table is the ability to take it OUTSIDE when doing a lot of routing. If the weather is good, I can make sawdust all over the drive way, and not in the shop. Cleanup is a lot easier with a leaf blower. Even though my table saw is on wheels, the idea of moving it in and out of the garage would be more trouble than it is worth.
If space is an issue, you can make a fold up router table. The base is a back with 2 gate leg panels, and the removable top locks everything in place. Pop the router plate out and set it on the shelf. There are also tons of other folding or quick set up designs on the web if you search for images of folding router tables