I often wonder why people compare Arduinos to 'Pis, that is a lot like comparing a desktop computer to a super computer...literally. Most Arduinos are roughly equivalent to a 1980's desktop computer, while many of the PI clones are more powerful than a 1980's Cray-2 supercomputer.
I like the Arduino mini-pros, I use those a lot for controlling different devices. Depending on what you're using it for, it's possible to power one of those of a coin cell battery for several months.
As for Pis, the ODoids are definitely cutting edge, probably the most powerful Pis available, but they are also relatively large and fairly expensive. Right now I'm partial to the NanoPi line from FreindlyArm.com. I'm using a NanoPi Neo2 for my email/web/dns/etc. server
What I really like about the Neo is the heatsink they use, especially when you have it mounted in the aluminum enclosure shown above. The heatsink presses on the enclosure, basically making the entire enclouser a heatsink. Because of this the CPU temperature runs less than 5C above ambient temperature. Power consumption is extremely low, even with the OLED display running it draws around 2/3 of a watt. I'm guessing without the display somewhere around 1/3-1/2 watt.
The other thing I really like is the price. The above setup, Neo2 with aluminum case and OLED display, was only $35 + shipping. With the NanoPi Neo (slightly less powerful than the Neo2) it's less than $30.