Well, unfortunately I can't tell you why you're having problems, but I can tell you what "finalising" does.
When you create (burn) a CD or DVD R/RW, finalising the disc causes the software to run a "closing" session. What this does is set the disc so that no further data can be written onto it. This should be a default setting for "recordable" media, but with re-writable media, if you do not finalise the disc, you can actually stick it back in later and add more data to it without overwriting what's already there.
A side-effect of not finalising a disc is that it can be incompatible with some disc drives, particularly older and/or cheaper ones. Whoever told you to finalise your RW discs was doing the right thing.
As for why they won't play in the other player (which I assume is not a recordable one) could be down to the fact that it may not be compatible with DVD+R discs. Some DVD players will only read from DVD-R (minus) discs (such as the PlayStation 2), so the best thing you could do there is check with the manufacturer for compatibility with the type of discs you're using.