The difference is only in whose devices Sync Pro is going to be used on.
Sync Home Pro license can be applied to person's own devices and used in non-commercial home environment for syncing personal files. The person may arrange devices into as many identities as they need - either get each device its own separate identities or link some of the devices with one identity. Also, the number of devices where this license can be applied is not limited.
Sync Family Pro license is actually the same as Home Pro but this license can be used on devices of up to 5 Family members - their own desktops, laptops, etc. Same as with Home Pro, each family member may keep a separate identity on each their device or link them with one identity, or link to that of other family members. The number of devices is also not limited.
License also does not impose any limits on syncing across family members' devices. Note: if family members don't need Pro features, they can use Sync Free on their devices. You, with your Sync Home Pro, will still be able to sync with them, they just won't have Pro features on their devices.
If you're the only person who's going to use Sync Pro on your own devices, Sync Home PRo license is enough, others may be ok with Sync Free.
If you have a home server, which is equally used by everyone in family, and you want to install Sync Pro here, Sync Home Pro license is enough.
But as soon as your family members - spouse, siblings,(grand)children, (grand)parents - want to use Sync Pro on their own personal devices, you need to get the license for them too - either a buy a separate Sync Home Pro, or Sync Family Pro so as to save on bulk purchase.
To use the Family Pro license, simply apply the purchased license key to all necessary devices. See here for more details about applying license key.