I never owned a Band 1 device, as it looked far too much like a version 1 product for my liking. It was big and clunky and didn't offer enough features that made me interested in wearing it.
The Band 2 has a curved screen and is more tailored-looking. It includes a barometer sensor and other updated ones, including its onboard GPS. Band 2, like Band 1, provides heart-rate monitoring; tracking for running, biking, golfing, cycling and more; and the ability to work with Windows Phone, Android and iPhone devices.
Band 2 supports Cortana integration. Currently Cortana support is only for English-speaking Windows Phone users with 8.1 or above, but Microsoft does plan to provide various degrees of Cortana integration for Band 2 users with Android phones and iPhones.
The Band 2 doesn't run any Windows 10 variant inside; it still has a firmware core. Because the Band 2 will connect with Windows Phones running Windows 10 Mobile and support various Microsoft services, it's considered part of the Windows 10 devices family by extension.
Band 2 is not waterproof, though it is water-resistant. Microsoft is quoting battery life for the new device at 48 hours, and says it will take 1.5 hours to fully charge Band 2.
The updated Band SDK includes support for creating and sending notifications to tiles. It also adds support for creation of PDF 417 barcodes with alpha/numeric data (maybe for boarding passes?).