Apparently Feanor devised a new alphabet just because he felt he could improve
on it. No other reason given that I've read. The older tengwar was invented by
Rumil and was called sarati. It was written vertically - maybe Feanor felt his
side-to-side method was better. It's very beautifully and reminds me of
sanscrit. (actually, the similarity is very fitting, since Sancscrit is more
ancient that the european-looking Feanorian tengwar) You can see it at this
site, Amanye Tenceli :
http://hem.passagen.se/mansb/at/