I am thinking of installing either Ubuntu 12.04 (LTS) or waiting for 13.04 to be released.
My question is: Will all the updates that appear in the new release also be made available to the LTS release? In other words, will all the improvements to Unity, all the new features etc., be available to 12.04 as well? Or should I go for 13.04 if I want all the newest features?
The LTS releases are expected to be very stable, suitable for offices, and "mission critical systems" like power plants. LTS gets all the "Important Security Updates" and any updates that fixes security issues.
LTS can be also updated with newer kernels, but it won't happen on it's own.
Typically, 12.04 LTS won't get the newest Unity features from 12.10, 13.04 etc. nor will it get newer versions of applications, like OpenOffice, unless such a feature or a newer application fixes a security hole.
Having said that, there are ways to get some of these into the LTS using the backport repositories and PPAs. In my opinion it is not worth the trouble unless you have some special circumstances preventing you to use the latest.
Normal releases on the other hand get the newest features and more recent applications like OpenOffice.
See What's the difference between a Long Term Support Release and a Normal Release? for more details.
To help answer your question about which version you should choose, my advice is the following:
Hope this helps
For new versions of unity with 12.04, it's not too simple (link). But for most other packages you can install from a PPA and then get the new version (link).