Alexa strengths
The largest catalogue of compatible smart home devices and skills, and generally the easiest starting point if you already own Amazon Echo speakers or Fire TV devices.
Google Home strengths
Stronger natural conversation and search integration, and tighter integration if your household already uses Android phones, Google Calendar or Nest cameras and thermostats.
Mixing the two
Many smart home devices work with both, so the bigger decision is usually which speaker ecosystem to standardise on for voice control, rather than being locked into one for every device.