What word means “almost a homophone”?

There is a linguistic term which refers to two or more words which have almost the same, but not quite the same sound. It has nothing to do with the spelling, however.

It's a term for the two words, as a pair. Beat [bit] and bit [bɪt] , for example, which differ only in their vowels - tense high front [i] and lax high front [ɪ] -- are said to be a Minimal Pair for the two sounds that differ.

Discovery of a minimal pair is evidence that the two sounds that differ are in contrast, not in Complementary Distribution. That means that native speakers have to routinely distinguish them because they are the only way they could tell the difference between the two words.

So they are therefore separate Phonemes. I.e, you can write them not just phonetically as [i] and [ɪ], but phonemically, as /i/ and /ɪ/. Knowing this is important to linguists (and occasionally interesting to others).

That's how we know there are around fourteen vowels in English, instead of the five Middle English "vowel letters" we're stuck with now -- in Middle English there were far fewer, but all of them came in two lengths, which were wiped out by the Great Vowel Shift. We have minimal pairs for all of them.

The simplest set of minimal pairs for English vowels that I know of in my American dialect is

beat /i/, bit /ɪ/, bait /e/, bet /ɛ/, bat /æ/, bought /ɔ/, but /ə/

bot /a/, boat /o/, put /ʊ/, boot /u/, bite /ay/, bout /aw/

There is no English word *[bʊt], but Luke and look are a minimal pair for them. /oy/ doesn't occur in this frame, but there are minimal pairs for all three diphthongs as well -- buy, boy, bough, for instance.

Yes, "quasi-homophones". "Quasi" means "seeming or seemingly; nearly". As a prefix it's generally hyphenated. It's pronunced /kwah-zee/.

I like "quasi-homophones," but maybe we should stick with "near homophones." Here's a list I made over a period of months. I didn't spend hours on this. Whenever I happen to think of a pair of near homophones, I add it to the list. (Some are nearer than others.)

ALMOST HOMOPHONES It’s windy today. No, it’s Thursday.  So am I; let’s have a beer. 

abdominal = abominable. anonymous = unanimous. appellation = Appalachian. culpable = capable. collusion = collision. condescension = condensation. cooperate = corroborate. curdle = cuddle. discussed = disgust. festival = vegetable. elicit = illicit. interface = interfaith. irreverent = irrelevant. melody = malady. militias = malicious. precedent = president. supplies = surprise. tempter = temperature. unaware = underwear. underserved = undeserved.