Good question. The most up-to-date DietPi version (v96) works just fine on Weezy and on Jessie. It’s just a piece of software.
Jessie also has some small bugs and some software will not work.
If you want to have Jessie, the best way is to reimage it on your Pi because the underlying software is completely different. You can also upgrade manually by following this little tutorial: http://raspberrypi.stackexchange.com/a/27859 (keep in mind that some programs could break because they are incompatible with Jessie).
About your other question, if you run:
It will show which version of Debian you are using.
Like this: REMOVED OUTDATED LINK
Regardless of your DietPi image (eg: Odroid / Jessie / Wheezy), dietpi-update will patch your specific system. Wheezy is still supported in DietPi, and, will continue to be patched along with all the DietPi images.
DietPi will never upgrade your installed distribution. So if your using the Wheezy image, your system will stay as Wheezy. This allows us to offer various distro types as images, and essentially, long term support for the old, yet stable previous distributions (eg: wheezy).
Most of the Jessie bugs we know of have been resolved (thanks to Pilovali’s testing), however, its still in beta. If you want to run Jessie, I would recommend downloading the DietPi Jessie image.