Depending on where you come from the understanding of which is which may differ completely. Some examples:
UK: lemon is yellow, lime is green
Portugal: limão is yellow, lima is green
Brasil: limão is green, lima is yellow, (the opposite of Portuguese!)
Mexico: limón is green, lima is sweet, smaller with a colour between the green lime and yellow lemon.
Estonia: Sidrun is yellow, laim is green
Germany: Zitrone is yellow, Limette is green.
Citrus fruit is said to have originated in South-East Asia and spread through Middle East to the Mediterranean (Greece, Sicily, Spain) and from Spain to Americas. That helps to understand the origin of the words.
Lime: origin in English is from French/Portuguese/Spanish and before that from Arabic līma, līm collectively meaning the fruits of citron kind.
Lemon in English originates from Middle English lymon coming from Spanish limón, Portuguese limão, Italian limone from Latin limō, before that Arabic laimūn from Persian līmūn.
In some languages the word root comes from the Latin citrus, (eg German Zitrone, Estonian sidrun coming from German Zitrone like many other German loan words). One of the types of yellow lemon is called Citron, perhaps that was widely grown and spred from old Romans to the Germanic areas.
It seems that in Latin and South America the colour of the fruit is often the other way round to the European concept. No wonder if in Mexico a European asking for some lemon would get the green fruit. Even better, a Portuguese in Brasil asking in their same language for limão thinking yellow and gets a green fruit.
Despite the fascinating etymological and cultural aspect limes and lemons are also an indispencable part of nutrition and cooking.
Both are rich in vitamins, especially vitamin C, calcium, potassium, magnesium, antioxydant flavonoids, and more.
The green limes known in Europe are Persian lime, also known as Tahiti lime or Mexican lime. The lime juice is the key ingredient in popular dishes like ceviche, aguachile, guacamole or in Margarita cocktails.
Key limes widely known in the US (from the Florida Keys) are smaller than the Persian/Mexican lime. Famous is key lime pie with condensed milk.
The yellow lemons are used in classic recipes like lemon tarte, lemon sorbet or in drinks like lemonade, in Europe gin and tonic is usually served with yellow lemons. The famous Italian lemon liqueur limoncello is made of Amalfi or Sorrento lemons that have exceptional aroma and flavcour and, I´ve read, are even protected under some legislation.
The juice of limes and lemons serve well as emulsifiers in salad sauces. Both go well with fish, shellfish and some chicken dishes.
Notes of interest:
The capital of Peru is Lima
Lemon and lime = Cockney „Crime“