These words are not really related
From Greek "syn" = same (like "synchronize" or "synonym")
Also from Greek "erg" = "energy"
So "synergy" means "same energy" or mutually reinforcing factors (usually positive but it is possible to talk of "negative synergies")
"Externalities" are outside forces i.e. factors outside a company, market or ecosystem
Externality - a side effect or consequence of an industrial or commercial activity that affects other parties without this being reflected in the cost of the goods or services involved, such as the pollination of surrounding crops by bees kept for honey.
So... if a beekeeper is keeping bees for honey but not charging the local farmers for pollination services, the free pollination is a positive externality for the farmers.
Similarly, if a company is discharging pollutants into the local water supply but not paying for the cleanup, it is a positive externality for the company and a negative externality for the local residents.