I have both. The SR-404 isn't a complete antithesis; both headphones are detailed and fast. The 404 has a much bigger soundstage, better extension top and bottom, and has a very different tonal balance, stressing bass and treble at the expense of the midrange. It's not lacking in bass at all, having more bass than my bass-light K340, but it doesn't have the same tactile bass slam as the K340.
I also have the A250, and honestly, the 404 sounds a lot like it, though it's not nearly as bright as the A250. The same problems with upper midrange prominence, thin midrange sound and slight shoutiness overall apply to the SR-404, though to a lesser degree. Of course, it's much better than the A250 in nearly all respects, and especially in treble quality (I can't stand the A250's strident, metallic treble, whereas the 404 is excellent in the highs).
If you're used to the absolutely pristine midrange of the K340, you will not like the 404. It's midrange is colored, and everything has a sort of electric glow and lustre. Tone color is very desaturated next to the K340, and instruments sound thinner and more metallic. The K340 also has more focused imaging - the 404 can sound somewhat blurred with softened leading edges, though it resolves a bit more detail than the K340. I suppose silver cables might help the 404 here.
I like the 404 on electronic music where it's midrange doesn't matter. I really don't like it on music with a lot of vocals, especially female vocals, which sound brittle and thin.
If you do go the 404 route match it up with tubes. The 007t is a very nice amp for it. You will want a warm punchy source that stresses the lower midrange as well.
I think the K340 has more potential in a high-end system due to a better tonal balance and less flaws overall. The 404 is technically a better headphone, but it's flaws make it less appealing as a listening experience. My opinion has shifted more times than Jersey weather when it comes to these headphones, and it simply comes down to being able to live with their flaws. On some music, I can do it, and on other music, I can't.
Last note... The 404 is very sensitive to warm-up. It's midrange coloration becomes less pronounced after about an hour of music at a normal listening volume. I don't think this is due to acclimatization to the headphones' coloration, since I usually let them play to themselves at a moderate volume for an hour before serious critical listening. When listening to them from scratch, the sound seems to change a good deal in the first hour becoming more balanced. Of course, this is a very subjective evaluation.