Who is this company and why would I buy a key from them instead of Yubico?

SoloKeys appear to use open source firmware, whereas Yubico's firmware is closed source since the Yubikey 4 I believe. Other than that, they seem about the same. SoloKeys is a lot younger and missing a couple of Yubikey Neo's bigger features (like PGP support).

OpenPGP is arriving, will be available via firmware upgrade.

https://github.com/solokeys/openpgp