Adding recordings to DVD works a bit different to adding sessions to a disc on a PC. When you insert a blank DVD, it creates an open session. As you make recordings to the DVD, these are added to the same session, just as when writing a bunch of files to DVD, each file is written in sequence to the current session. Once you finalise the disc, the recorder writes the lead-in and lead-out, just as the PC software writes the lead-in & out-out once it finishes writing the fileset to disc. However, unlike writing data to a multisession disc on a PC, it is not possible to start a second session on a recordable DVD-R or DVD+R in a DVD recorder, which means that once the disc is finalised, no further recordings can be made to it.
The unfinalise feature you have can actually be handy for for DVD-RW's should you need to be able to play it in a DVD-RW VR compatible player or PC, but need to record to it again later. Unfortunately as I mentioned, DVD-RWs recorded in the VR format will only play back in VR compatible DVD players. As far as I'm aware of, only certain PC DVD playback software will play these also, so it is nice to see that you have suitable software.
As far as I'm aware of, DVD+RW discs are only written in the VR format, which means that there will be no choice given when you load a blank DVD+RW disc. However, unlike DVD-RW VR discs, the DVD+RW VR recording format is designed to be compatible with existing DVD players and should not need finalisation either to play elsewhere. For example, you should be able to record to a DVD+RW disc, take it out, place the disc in a PC or other player and start playing it.