What to Know
Biweekly and bimonthly can mean the same thing because of the prefix bi-, which here can mean “occurring every two” or “occurring twice in.” Therefore, biweekly can be “twice in a week” or “every other week.” Bimonthly can also mean "every other week" if it's twice in a month, or it can mean “every other month.”
Look up the adjective biweekly in this dictionary and you will see it defined as "occurring every two weeks" AND as "occurring twice a week." Similarly, the adjective bimonthly is defined as "occurring every two months" AND as "occurring twice a month."
For this, we are sorry. But we don't mean "sorry" in the sense that we feel penitence; we are not to blame. We mean "sorry" in the sense that we feel a kind of sorrow aroused by circumstances beyond our control or power to repair.
For, as anyone who pays attention to our work surely recognizes, we are at the mercy of the language. We diligently record the English lexicon in both its measured expansions and its wild proliferations, and any insistence by us that it favor the former over the latter is as whispers into a gale. Biweekly and bimonthly each have a pair of meanings that are unhelpfully at odds with one another. Those meanings exist, and we cannot ignore them.
The problem lies in the prefix bi-: it means (among other things) both "coming or occurring every two," AND "coming or occurring two times." This too is a long-established fact that we cannot ignore. English is sometimes simply obstreperous.
However intractable the English language may at times be, it can be helpful to remember the many cases in which the language is not ambiguous, in which its offerings include words that refer to precise gradations or fine distinctions.
One such case, sort of, is very similar to the cases at hand: the language offers us biannual for "twice a year" and biennial for "every two years." This is useful and elegant, but, alas, also frequently botched, with biannual so frequently used to mean "every two years" that we've had to enter that meaning in our dictionaries. But here another solution is readily available: skip biannual altogether and use in its place the common semiannual.
Ah, semi-! Just as a semicircle cuts a circle in half, so too does the prefix semi- semantically cut what it is affixed to in half: semiweekly means unambiguously two times per week; semimonthly means two times per month; semiannual means two times per year. It's an excellent option, and one that many writers seem to embrace; we most often see bimonthly and biweekly reserved for their "every two" meanings.
Another solution is to avoid the bi- compounds altogether and come right out with it: "twice a week," "every other month."
As writer or speaker, you can choose to avoid ambiguity by using a semi- compound when you mean "two times per," or by using a phrase instead of a single word. In fact, so fraught is this territory that you'd do well to make sure that your context explains just what you mean when you use any of these bi- compounds: "Employees are paid bimonthly, on the first and third Fridays of the month."
But what about when you're at the mercy of English as it's wielded by others? Well, when you are a reader or listener, we're afraid the best you can do is to approach biweekly and bimonthly with a bit of side-eye—and perhaps the kind of sorrow aroused by circumstances beyond your control or power to repair.