I have both bikes. They're both really good machines and either would be a workable and reasonable step up from an EX250.
The Katana is heavier than the SV, and you feel it a bit in city riding, but other than tight, heavy traffic commuting the weight of the Katana never bothers me compared to the SV. SV is a bit more enjoyable in heavy traffic, but the clutch is a lot heavier, so stop & go is kind of a wash between the two bikes. Also, when the rad fans turn on on the SV, you get a mean (in the summer) or welcome (in the cooler weather) blast of heat.
This "down on power" crap is a bunch of bull for the most part, the Katana's aero gives it a bit higher top speed, and it's more comfortable at speed. I expect that these comments are coming from people who are used to riding a twin and don't know how to get the most from an I4. From a standstill the SV is quicker, but WOT stoplight-to-stoplight riding is really not my concern.
This spring I rode my SV650 (naked) 1000 miles from Toronto, ON to Asheville, NC, then about 500-600 miles in the mountains there, then did a track day at Carolina Motorsports Park, then rode 1000 miles back home. Interestingly, I did exactly the same trip last spring on my Katana, so here are my thoughts:
1) Distance riding: Significant preference for the Katana. The SV is capable on the highway, but you've got better weather protection on the Kat. I have stock seats on both and found the Kat marginally more comfortable in the butt department. Seating position and ergos are almost identical (with the S clip-ons).
2) Mountain twisties: Slight edge to the SV. For really tight technical stuff, the SV has it. For flowing sweepers (which is my favorite), the Katana is at least as good as the SV.
3) Track day: HUGE props to the SV. I had an awesome time on my SV at this track. I can't compare it to the Katana, because I haven't and wouldn't do a track day on the Kat.
I consider the Katana more reliable than the SV, and when it does require maintenance, it's about 100x easier to work on for anything engine-related. On the Katana, you don't have to pull the rad to get at sparkplugs, or remove the shock to change the exhaust. Nor do you have to remove the carbs to connect the vacuum hose to do a carb sync. For chassis/suspension/brakes stuff, the two bikes are the same in terms of maintenance.
RandyO's 135k mile SV is kinda legendary, but nobody would be surprised to see that many miles on one of the Suzuki a/o cooled I4 motors. I genuinely consider this the greatest motorcycle engine ever built. With the Kat 600, you've got ~80hp coming out of an engine architecture and structure that is reliable at 150hp. In the Kat, it's way understressed and it just runs and runs and runs.
Personally I love the looks of the Katana, but that's a purely personal thing, so decide for yourself.
Both bikes have OK brakes but pretty crap-tastic suspension. An aftermarket shock and a set of Race Tech cartridge emulators & springs should be considered absolutely mandatory for either bike. I can not stress enough how important it is to upgrade the front suspension (not just springs, spacers, or an oil change!) to get the most out of either of these bikes!!!
They're both fun, good-looking, fast enough, and cheap bikes. If you enjoy wrenching on your own stuff, the Katana has clear advantages here. If a lighter, narrower bike is more important to you, and you want to do some time on the track, then the SV's your machine.
Really I don't think you can go wrong either way, and you certainly don't have to consider one as a stepping stone to the other.