Why does it matter who beat who by a few years in obscure mathematical discoveries? Because we're a science site so we like obscure historical factoids. And he lived in feudal Japan, where samurai were all the rage and the shogun's isolationist policies meant there was little information coming from outside, so discoveries were a lot more difficult. In addition, mathematics was primarily regarded as an elegant pasttime and not a legitimate pursuit, sort of like tea ceremonies and calligraphy. Other than simple bookkeeping, no one really did math. So the son of a samuari choosing to go into math was really something. It's not easy being the black sheep of the family when papa wields a katana.
That makes coming up with the kernel of a general theory of elimination 150 years before Bézout, when Japan was far behind China and the west, all the more spectacular - in kanji there were not even symbols for parentheses, equality, or division before him. Imagine trying to do advanced math in a language that doesn't have a written concept of division.
Bonus: He also calculated pi out to the tenth
decimal place also by using an 'accelerating
the rate of convergence of a sequence'
method formalized by Alexander Aitken in
the 20th century.
So if he did all those great things, why would his work have fallen away? Believe it or not, there were competitive schools of mathematics in Japan. You think arguments over metric versus standard can get ugly? Imagine being in a country where multiple mathematical systems were undercutting each other. Since wasan, his mathematics, was not good for astronomy or surveying land, they said, what good was it?
Plus, Seki's work was too darn hard for descendants - literally requiring calculations beyond ordinary human ability. At some point there are no worlds left to conquer that can be conquered without a computer; elimination theory advancements require some real number crunching.
Over 160 years after his death, wasan and the other indigenous systems were overthrown anyway, when feudal Japan became imperial again and allowed relations with the west; including mathematics.
But it's always nice to learn about a self-made, self-educated man of learning; I'd even be willing to hear arguments that Seki's yenri or "circle principle", was on a par with Newton's calculus of fluxions.