There are several discussions in the archive (Suche in allen Foren, oben links), but at a quick glance, none of them seemed all that helpful. I could have sworn there was something better somewhere, but ...
'Company' is the normal, general word. When in doubt, use it, especially when there is no qualifying adjective before it to limit its meaning.
Our company has 250 employees.
I would like to apply for a job with your company.
He accepted a position with another company.
How many companies are listed on the stock exchange?
She works for a company that manufactures home appliances.
Do you think the company will negotiate better terms with the union?
Also, as was pointed out in the other threads, only 'company' can be used as an adjective:
Sales representatives are issued a company car.
Company policy is to use our logo on all printed communications.
'Firm' is more formal/elevated and often more specialized. It's more often used following a particular qualifying adjective than by itself: a __________ firm.
It's most common in reference to groups of professional partners that provide specialized white-collar business services. One very standard collocation where you would not use 'company' is law firm. Somewhat less clear-cut are, for example, accounting firm and (even less so) engineering firm, for which 'company' is a more thinkable but probably still less common alternative. However, I don't think we would call an association of, say, doctors or dentists a firm, but rather something like a group or partnership.
People also sometimes use 'firm' just to lend a touch of elegance in more formal or promotional contexts. So, for example, manufacturing firm may sound a bit more impressive in writing or advertising than just 'manufacturer' or 'manufacturing company,' which would be more common in ordinary conversation.
Hope some of that helps. (-: