The two competing technologies use different formats. No single company "owns" DVD and both technologies have their "champions".
DVD-R/RW was developed by Pioneer. Based on CD-RW technology, it uses a similar pitch of the helix, mark length of the 'burn' for data, and rotation control. DVD-R/RW is supported by the DVD Forum, an industry-wide group of hardware and software developers, and computer peripheral manufacturers. The DVD-R format has been standardized in ECMA-279 by the Forum, but this is a private standard, not an 'industry' ISO standard like the CD-R/RW Red Book or Orange Book standard.
DVD+R/RW is also based on CD-RW technology. DVD+R/RW is supported by Sony, Philips, HP, Dell, Ricoh, Yamaha, and others, and has recently been endorsed by Microsoft. DVD+R/RW is not supported by the DVD Forum, but the Forum has no power to set industry standards, so it becomes a market-driven issue.
DVD+R is a dvd disc that allows multiple layers for one disc where as dvd-r only allows one layer. They will not compete to become the de Facto standard, because they are both here to stay. Multi layer DVD+R can allow extra capacity per disc than DVD-R hence its high cost!