What is the difference between "compañía" and "empresa"?

A "Compañía" is a group of people aiming for a common purpose, oftenly a commercial one. Examples: Compañía de luz, compañía naviera, compañía de bomberos, compañía de actores, compañía de seguros, compañía forestal.

A "Empresa" is an organization with profit-making objectives.

In most cases, both are used interchangeably, but there are some interesting collocations such as: "Compañía de seguros" which is preferred instead of "Empresa de seguros".

I'd recommend you to employ "Empresa" since it is more widely used.

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What's the difference between an enterprise and a company?

They can be synonymous to mean a business, but compañía can also mean company as in "I have company over". You can use either one, I use empresa most of the time, personally.

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Empresa proviene de emprender. Compañía viene de acompañarse o asociarse. Pero en el mundo de los negocios ambos términos pueden significar lo mismo.

Según RAE:

Compañía: Sociedad o junta de varias personas unidas para un mismo fin, frecuentemente mercantil.

Empresa: Unidad de organización dedicada a actividades industriales, mercantiles o de prestación de servicios con fines lucrativos.

Como podéis ver, son bien similares ambas definiciones.

Terms in English that can be used for specific commercial entities:

enterprise business company

firm (usually a partnership of professionals but not necessarily)

concern (very general)

outfit (very informal and colloquial)

association

corporation (legally defined type of company)

partnership (legally defined type of enterprise)

house (rare but still used for certain types of businesses)

Indeed, what is the difference between an entrepreneur and an undertaker?