There are two main systems of Romaja (English romanizations of the phonetic Korean script of Hangeul). The first, Seoul Revised, is the standard set by the South Korean government, the majority system of Romaja, and the simplest one to use in my opinion. In Revised (or Standard), the title for teacher is Sabeomnim.
The second, McCune-Reischaur was invented by two American scholars, and is based on a system of phonetic pronounciation rather than transliteration. This makes it a more accurate read if you understand phonetic symbols, but difficult to type and write with. In MR, teacher is sapŏmnim.
The thing to remember is that in Korean, there is no difference. This sound is represented by 사범님. Hangeul is a phonetic script, just like English, and a Korean, hearing these sounds, will think these characters, and vice versa. Hangeul, being one of the most recently invented phonetic scripts, is actually very easy to learn. Unlike English, which has multiple sounds for one syllable (how many phonetic sounds match with the letter "a"), Hangeul has one symbol for one sound. So, while there may be 72 symbolic combinations, there is only one sound to go with each one. I could take anyone on this board and have them reading Hangeul easily in two weeks maximum (note: reading is one thing, comprehending is another...)
In addition to the term Sabeomnim (I prefer Revised) being written 사범님 in Hangeul, it is actually based off of a Hanja term. Hanja is the Korean word for Chinese characters, which were used before the relatively recent invention of the Korean phonetic script. Chinese characters represent whole words or concepts, unlike phonetic systems, which only represent sounds. This differentiates Hangeul from Chinese, in that Hangeul is phonetic, like English. The English term teacher carries no weight as an individual character, but is rather made up of syllabic phonetic clusters which we recognize as the sound meaning "teacher". Chinese characters rather convey a concept which is matched with a sound. You must recognize the character itself in order to understand the written language, you cannot just "sound it out" like you could in English or phonetic Hangeul.
Sabeom is a term made up of two Hanja characters. The first, 師, represents the idea of someone of has mastered a given skill set, and is thus able to teach it. In Korean, this character is pronounced 사, Sa.
The second character, 範, pronounced Beom in phonetic Korean and written 범 (Beom), represents the ideal of a mean or rule, a model or pattern, a mold. In this context, a sabeom is not just a teacher, but rather, the very pattern from which the idea of teacher should be taken. The sabeom is the person other teachers look to, the role model, the idea of what a teacher wants to be when they think of the term teacher.
It should be noted that this term does not actually mean teacher, but rather was adopted from the Japanese martial arts term shihan, and was not found in pre-Occupation Korean martial arts. 師範 is Shihan in Japanese. This term is not a native Korean martial arts term, but rather an adopted one. That is not to disparage its use, as it has valid philosophical and linguistic meaning. However, understanding the etymology of the vocabulary involved in one's art increases the ability to understand and appreciate one's art.
As an example, this term is not used in the system I was trained in. We use the term 師父, 사부, Sabu, which literally means Master/Father. This implies a familial relationship between practitioners. In Mandarin Chinese, the necessary Hanja would be pronounced Sifu. See where I'm going with this? Of course, in a Korean system, this term is written and pronounced phonetically as 사부님...Sabunim...
님/nim is an honorific following titles. It has no Chinese character because it is not Chinese, but rather Korean. It is used only when spoken. I may write of my teacher as being a sabu or sabeom, but only when speaking of them do I call them sabunim or sabeomnim.
So, where does this leave us with alternate Korean spellings of martial arts terms?
Sa Bum Nym and the like arise out of two main sources.
The first is Korean teachers themselves. Expertise in martial arts and fluency in Korean does not necessarily equal an ability in linguistic translation and phonetic transliteration. Thus, many martial arts schools are left with their founder's "best guesses" on how to translate and transliterate the necessary vocabulary. Move a few generations down the line of transmission, and much like the childhood game of telephone, the final results can be a bit humorous if one actually intends to converse with Korean nationals on the subject. Outside of that however, it is not a major issue.
The second source is Westerners making their "best guesses" as well. Here we find the various source phonetics coloring the transliteration, much as we would with Korean teachers. We like to think we hear things as they are, but when it comes to spoken language, we instinctively hear things as we are trained. If French was our mother tongue, that is the phonetic set we think/hear in, English, Korean, etc. It takes a great deal of diligence and linguistic training to overcome this, and frankly, very few (myself included) can.
The fact is that there are sounds in the Korean language that are not phonetically present in English. Vice versa, there are sounds present in English which do not translate well into Korean. In English, we have no phonetic equivalent for the sound 으. In Revised, this is written as Eu. It sounds like the sound you may make while shivering from extreme cold. In English, we distinguish between the "L" sound and the "R" sound. Korean does not. ㄹ sounds like L at one moment, R at another, and sometimes a rolled bastardization of both.
Those who brought the KMA to Western ears did the best they could, be they Westerners or Koreans. We do not need to cling to their sense of transliteration out of tradition, but we can look back and see what enormous hurdles they had to overcome to introduce their teachings to an entirely different culture.