Why did Peter Jackson direct the Lord of the Rings first and then The Hobbit?

Most of this with appropriate additional source attribution can be found on the wikipedia pages of The Hobbit and Lord of the Rings.

When Jackson first started on this (1995) he wanted to make 3 movies, one based on The Hobbit, two based on Lord of the Rings. However, it turned out at that point that the production rights and distribution rights on the Hobbit were owned by different parties. To the best of my knowledge they decided to continue solely with Lord Of The Rings afterwards. This project went through quite some production hell (going from 2 movies to 1 to almost none to 3). Only after the extreme success of those movies the team started looking at the Hobbit again. That also took some time because first those rights had to be obtained and secondly there were some legal struggles between Jackson and New Line Cinemas. However, eventually everything was settled and the films were also made.

Now the following is more subjective, but as you mentioned the digitial effects specifically, I want to comment on that.

People tends to overestimate the role of technology and technology evolution. First of all, technology evolution is relatively limited, most computer rendering methods are already quite old. For example, ray tracing was developed late 60's/early 70's. Recent evolution only provides a few things on top of that: more processing power (and memory) and improved tools. In the end it depends at least as much on the skill of the artist how good the results look, and together with that also on how much time they're given.

The first is for cinema actually less of an issue, as there is no need to render frames in real time, so whether one has to wait 1minute or 1 day for a final, production frame to render is not important (for big budget productions). Same goes for memory, it mainly makes it go faster, but if you don't have enough RAM nothing stops you from rendering from (a set of) SDD/HDD's.

The second is more important, because better tools (should) save time. Suppose for example you are working on a render of a forest. If you have a tool that's optimized to generate quite realistic vegetation as a start, you'll be up to speed a lot faster than when you have to design everything from scratch. If the tool also provides a way to model a breeze through that vegetation, then your animation task also becomes easier, etc. etc. .

However, for LOTR specifically Weta Digital created those tools themselves, most famous example (also mentioned in the question): they needed to model huge armies so they made a tool for that called MASSIVE, which was later actually reused to do some of the work on Avatar.

Finally, don't underestimate how much 'special effects' in Lord of the Rings and even more recent movies is done using good old props. For Lord of the rings this was again done by Weta. Here CGI technology evolution would again have little to no impact.

So in the end it seems unlikely that the more than 10 years of technological advantage would have made a huge difference for lord of the rings. Some for sure, but probably not worth the wait.