You probably don't need to know Greek mythology. I mean, it's not very likely that you'll be in a life or death situation where you'll have to veer your spaceship away from the Titan (a) and King of the Gods (b) planets and back towards the Love (c), War (d), and Messenger (e) deities in order to find your way back to Earth. Nor will it make very much difference if you fail to recognize the mythological figures behind the name of your car (Saturn or Mercury). However, Greco-Roman mythology is pervasive in Western culture and you probably already know a lot about it:
The love goddess Venus, whose name is synonymous with beauty, is featured in song and art. Her name was lent to what used to be called social disease. Adonis, one of her lovers, is synonymous with male beauty. The narcissus flower was originally a vain young man. The laurel was a young nymph who preferred to be turned into a tree to the embraces of Apollo. The space mission Apollo is named for the god of music and prophecy. There is a petroleum company whose logo is the winged horse Pegasus. An automobile muffler company is named for the original man with the golden touch (f). A moving company is named for the Titan who was punished by having to carry the weight of the world on his shoulder (g). One brand of running shoes was named for the goddess of victory (h). A sink cleanser was named for a second-best Greek hero in the Trojan War (i) after Achilles had died. The number one hero gave his name to the word for a long, difficult trip or odyssey. Odysseus also devised the original gift that gave us the expression "beware of Greeks bearing gifts" (Timeo Danaos et dona ferentes). A chocolate candy company is named for the Roman god of war (d). Cereal is named for the Roman goddess of grain (j). The panic button is named for a son of Hermes (k). The list goes on and on.
It may not make an appreciable difference to your quality of life, but knowing something about Roman and Greek mythology will give you insight into our cultural heritage, an understanding of the naming of the space and exploration missions, and it may help you solve a crossword or two.