One example that I learned relatively early in my entry into the Spanish language is that "embarrass" does not mean the same thing as "embarazar"; and therefore "I am embarrassed" does not translate into "Estoy embarazado." Since the word "embarazar" means "to be pregnant", the past participle (i.e., "adjective" form) of "embarazado" (in the masculine) not only isn't the correct translation of "embarrassed", but it is also a physiological impossibility outside the film Twins or the case of seahorses.
Another example of a false cognate between English and Spanish is between the English word bigot and the Spanish word bigote. Now a mustache (bigote) does not a bigot make, and indeed, the use of bigote in Spanish idioms is very different in meaning than bigot. For example, "Hombre de bigote" means more than "man of mustache," having the additional meaning of a "spirited or vigorous man". Looking at the word origin of the word "bigot" in English doesn't really help much, either:
1590s, from Fr. bigot (12c.), in O.Fr. "sanctimonious;" supposedly a derogatory name for Normans, the old theory (not universally accepted) being that it springs from their frequent use of O.E. oath bi God. Plausible, since the Eng. were known as goddamns in Joan of Arc's France, and during World War I Americans serving in France were said to be known as les sommobiches (see also son of a bitch). But the earliest French use of the word (12c.) is as the name of a people apparently in southern Gaul (which led to the now-doubtful, on phonetic grounds, theory that the word comes from Visigoth). Sp. bigote "mustache" also has been proposed as a source, though the sense is not adequately explained. The earliest English sense is of "religious hypocrite," especially a female one, and may have been influenced by beguine. Sense extended 1680s to other than religious opinions.
So, are bigot and bigote derived from the same word root? Who knows. However, there are others that definitely are. One example with a twist is the Spanish word recurso, which can be used in place of two different English words: recourse and resource. Why might this be the case? I mean, it is possible to make a rationalization for how recourse and resource can be related to each other, but looking at the word roots, one can see that they have different ones in Latin (recurrere and resurgere, respectively). Therefore, it seems like this is a linguistic "evolutionary conversion" in the case of Spanish, and, indeed, if one were to look a little further, one notes that the verb form of the recourse equivalent is recurrir (which is very similar to the Latin recurrere, no?) while the resource equivalent has no verb form (that I could find). Therefore, in translating from Spanish to English, one has the potential of falling into a false cognate problem, but not from English to Spanish. (Although I wonder if the direct translation of the phrase, "a recourse to resources" translates properly as, "un recurso a recursos.")
Sometimes, the false cognate is a misinterpretation of a word or grammar structure, as in the case of the Spanish-used "walking closet" to denote a "walk-in closet". While this might easily be done across languages, it is also not uncommonly done in the same language: "all intensive purposes" vs. "all intents and purposes", and one can find a list of many common misconstructions likely due to misinterpreting an overheard phrase).
Other languages obviously share false cognates, and it seems obvious that the further apart the languages, the greater the chance that two words that sound the same/similar between languages are likely to be false cognates (not counting borrowed words, of course). And this is true of words between Japanese and Spanish. Just for example, the Spanish word ajo (garlic) is a direct homophone of the Japanese word aho (idiot, jerk); the Spanish word vaca (cow) to the Japanese baka (stupid); etc. However, strangely, there are also some times where, through, coincidence, two words happen to have the same meaning, like the Spanish verb mirar (to see) is similar to the Japanese verb miru (to see/to watch).
As a side note, however, the Japanese, when historically importing Portuguese words (and later English, German, and Spanish words) into their lexicon, oftentimes went the extra distance of actually using homophonic kanji that had a similar meaning to the transliterated word. Therefore, the old Japanese word for a man's suit is sebiro (written with the characters for "wide" and "back"), which invokes the famous tailors' street in London, Saville Row, and the cut of suit coats that they made which seemed to emphasize wide shoulders on the wearer. To that end, it isn't always easy to tell if a word is borrowed explicitly (such as the Japanese word pan, which was borrowed whole-cloth from Portuguese) or not (such as the Japanese word tokei ("time" + "total" = clock), which could be derived from the Spanish word toque (3rd person singular present form of the verb tocar, to touch, or chime)).
Finally, I am finding it interesting to look at how acronyms can also suffer the consequence of something akin to a false cognate. One good example of how an acronym can be used to create an interesting cognate is the term LASSO, which (in statistics) "is a shrinkage and selection method for linear regression" (a very good parallel of what a lariat also does). However, I recently came across a paper that used the acronym DIO (diet-induced obesity). It seems to me that the person who coined the phrase either didn't know the meaning of the Word (yes, that's supposed to be a joke), or was being purposefully ironic. I'm leaning toward the second option, especially since it is quite true that an increasing number of academic paper authors are coming from the non-European languages, and might not always appreciate the multiple meanings of words. (Still, though, not everyone is prone to this on all occasions; like the student who came to me to figure out how to make a better acronym to describe his method of Five Unit Kinship Description.)