Very simple: Verizon Cloud is hosted by Verizon, iCloud is hosted by Apple. Both Verizon Cloud and iCloud backup most of the same content (contacts, photos, songs, videos, etc.) Both provide 5 GB of storage for free, with the option to upgrade. If you have a More Everything plan, you receive 25 GB free.
Verizon Cloud says it can backup text messages which iCloud does not. However, iCloud has some features that Verizon Cloud doesn't have. I'd recommend using one has a primary, and the other as a backup. I use Verizon Cloud has a backup incase I ever decide to change my phone.
Verizon Cloud: Verizon Cloud FAQs | Verizon Wireless