Functionally, they are not perceptibly different.
Let's shuffle things up.
Imagine that VHS tapes (The old movies) were made for Video Cassette Players (VCP) instead of Video Cassette Recorders (VCR). Then, lets say a manufacturer decides to make a VCP that can also record to "VHS-R" recordable VHS tapes, creating the first VCR. The manufacturer decides to allow other manufacturers to create VCRs that use the same recordable tapes for a price. These recordable VHS-R tapes will play back on a standard VCP. But they aren't the only people that can make VCRs and recordable VHS tapes compatable with older VCPs. So they make their own VHS+R VCR and tapes because if they made a VHS-R compatible product they would legally have to pay the original manufacturer. It functions identically except that because the recorder is from another manufacturer it requires that you buy recordable tapes from another manufacturer. This new manufacturer's technology is quickly adopted by the rest of the industry because they no longer have to pay the first company (Competition lowers prices). Once both companies are on par with price and market penetration (Public acceptance), you start to see dual-format VHS+/-R VCRs.
It's like you had a VCR that can play all VHS, VHS-R and VHS+R tapes but can only record VHS+R tapes. If your friend has one that can play all VHS, VHS-R and VHS+R tapes but can only record VHS-R tapes, you can still play what he has recorded. The only concern you have is which recordable tape to buy in the store for your VCR. If you have a newer VCR that can play all VHS, VHS-R and VHS+R tapes and can record to both tape formats (VHS+/-R), you don't even have that to worry about.
The R and RW thing is just like it's always been for CDs. You won't find a DVD-R drive that doesn't record to DVD-RW, so don't worry about that!