Both are based on the same hardware: a Bosch System In a Package (SIP) that includes 9 DOF sensors and an ARM M0 processor. But the BNO055 has firmware developed by Bosch whereas the BNO080's firmware is entirely written by Hillcrest Labs. With the different firmware comes different sensor fusion algorithms, different power performance, different interface protocols, support for additional sensors outside the SIP, etc. In short, everything is different except the silicon. :-)
The BNO080 supports a very rich set of sensor reports, to read both the raw sensors and the various fused data products. One thing our chip excels at, that is probably of interest here, is providing accurate heading (and, more generally, 3D attitude).
There are two interface modes: one is super simple (RVC mode) and the other is fairly complex (SHTP). In RVC mode, the BNO080 simply transmits a data report every 10ms (100Hz) that contains roll, pitch, yaw and accel values. (Based on Accel and Gyro sensors, not Mag). Just power it up and start reading the data with a UART.
SHTP was designed for low latency and high data rates (think VR) as well as low power wake-on-event functionality (think cell phones). It runs over a UART, I2C or SPI interface and supports interrupts and sleep capabilities. Hillcrest publishes an interface library with example code on github:
https://github.com/hcrest to help people interface with our chips using SHTP.
Dave