Apparently another technology, Kinemacolor, was invented first in the UK, but it never caught on in Hollywood. Kinemacolor was expensive, and had some other glitches involved, but the fact that it was first on the color movie scene doesn't seem to have saved it. What are some of the differences between the two that might have contributed to Technicolor prvailing over Kinemacolor? Was Technicolor that much better, and easier to use?
The two technologies never directly competed (as the question implies). The last Kinemacolor film was made in 1914, while the first Technicolor film wasn't made until 1917, and it doesn't appear to have entered serious usage until the 1920's.
So the real question we are left with is why Technicolor succeeded, where the earlier technology failed.
The main clue I see in the Kinemacolor wiki page was that the company never made money (despite installing projectors in several hundred theaters), and some mention of the projectors being expensive:
However, the company was never a success, partly due to the expense of installing special Kinemacolor projectors in cinemas. Also, the process suffered from "fringing" and "haloing" of the images, an insoluble problem as long as Kinemacolor remained a successive frame process.
The Technicolor folks, coming along a decade later as they did, would have been in a position to learn from the financial mistakes of their predecessors. In particular, it appears they put a priority in coming up with a system that didn't require all those expensive projectors:
The difference was that the two-component negative was now used to produce a subtractive color print. Because the colors were physically present in the print, no special projection equipment was required and the correct registration of the two images did not depend on the skill of the projectionist.