Ughh this is gonna be another lamo kawasuki verse, yamaha thread.
I wish one of these threads would contain meaningful information about hull designs and etc that make it better at free style. The only thing I've ever read is about engines and you can put a bigger engine in anything.
That's not very useful so don't waste your time searching.
I have gathered is that the Yamaha has a flatter bottom so it steers more with the back, and the Kawis steer more from the front. That means, the SJ slides and the Kawi carves.
The yamahas are also said to be more reliable.
The engine discussion is negligable outside of reliability. Sure the SJ has a bigger engine than a SX (Only if you buy a 1994 or newer, so if you went with a same year Kawi you'd probably have a 750 and that means the Kawi still has a bigger engine). You can put a bigger engine in anything. Comparing apples to apples, Kawis have larger engines stock (generally, remember to compare years, an 89 SJ has the same size engine as an 89 SX. The 2008 SJ has a 701 and the 2008 Kawi is an 800, Kawi = bigger).
Yamaha is the preferred brand so there are more after market parts availible to you, especially used, which is a huge plus.
Saying "I like the SJ better because it's better" is not really that helpful.
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To directly answer your question to the best of my limited ability.
The SJ has a flatter bottom which gives you better handling for freestyle. There are more aftermarket freestyle options avail to you. More parts are avil used for the budget concious. The yamaha is less lekely to break down so you have less maintance, if you go with a SJ with a 701 in it you will have a bigger engine, but that requires a larger budget, so if you want an older SN, you will have the same size engine as the SX. The difference would be that the SJ has more mods avail so you will have a better chance of buying a used pipe or something to get a bit more out of the same size engine.