There are (almost) no sets of words in any language that mean exactly the same in all circumstances. If such a thing happens, usually one word stops being used.
The verb fathom is a nice one. If we look at etymonline.com, we find this:
Old English fæðmian "to embrace, surround, envelop;" see fathom (n.). The meaning "take soundings" is from c.1600; its figurative sense of "get to the bottom of, understand" is 1620s. Related: Fathomed; fathoming.
I know the related word vadem in my native Dutch as relating to measuring the depth of water under a ship. This is the "take soundings" meaning in the etymonline text. Taking soundings is a way to measure the depth of water.
From that depth of water (you have to get to the bottom to measure that depth, with sound or with a stick), the meaning "get to the bottom of something" evolved.
In a way, that means "to understand something", but the implication is that you are talking about something reasonably complicated, that needs some work to understand.
You can say that you fathom the implications of the General Theory of Relativity (although many would doubt you are right), but if I tell you that you should look before crossing the street, you would not "fathom" that.
So if you mean understand in the sense of getting to the bottom of a complex problem, it means the same as fathom. But understand as in "do you understand what I said?" or "I don't understand this question" cannot be replaced by fathom.
Laure mentions two other expressions in the comment and I think they are great to compare with.
In the sense of getting to the bottom of something, figuring something out is a good alternative to fathom.
In the sense of probing (what etymonline mentions as the literal taking soundings),
estimating also is a possible meaning of fathom:
I could not fathom the depth of his hate.
In that sentence, fathom implies indeed that I can not estimate that depth. I could again use understand here, but it would not convey the (exact) same meaning.
So, use fathom when there is a definite sense of a process involving estimating something or figuring something out, or coming to understand something. Generally, do not use it when you describe the (static) situation of "having understanding of something".
Overall, do not use it too often anyway, because it has an antiquated feel to it for many people. Fathoming the depth of an emotion I would almost see as a fixed expression by now, and it works well because of the link to "measuring depth".
I would characterize the use of "recites" in the linked post as technically not incorrect, but very unnatural and antiquated.
That said, definition 4b from the OED does not support their claim that their use is in keeping with English usage of these words -- outside of legal contexts, it is not. That definition supports examples like "I recite this notice," but being able to use "recite" as a transitive verb does not make it acceptable in this context. "Read" may have a similar meaning to recite in contexts like "I read this notice," but unlike "recite", "read" is acceptable as an unaccusative verb. Many verbs, like "read", can be used as intransitives with the patient as the subject, but this is not typically the case for "recite," at least not in modern non-technical usage. The examples under definition 4b confirm this: they are examples in which an agent (in this case, a person) recites a text, not in which that text recites its contents.
However, there are indeed attested examples in the OED (which I also have access to) of "recite" being used as they claim it's used.
The relevant definition here is definition 1, which describes a definition of "recite" specific to the legal field.
The "also with clause as object" is what would describe the linked author's post, as that's clearly how they're using "recites" there. However, you'll notice that this definition specifies the Law domain, and that the examples from the 20th century are all from law reports and journals -- this is because using "recite" in this manner is really only accepted in the legal field, where language change occurs very differently and much more slowly than it does in the language used elsewhere (such as on Stack Exchange).
This is no doubt why the original post sounds so utterly wrong to some native English speakers -- because this usage is antiquated and unnatural outside of a very specific legal register (one most of us are not familiar with) and was used outside of the social context in which that register is appropriate.
This is why it is important for learners to pay attention to the sources of quotes like these when they're included in dictionaries, particularly in dictionaries like the OED, in which the examples can come from a wide variety of sources written in a huge variety of different registers. Using the right register for the context is a huge part of making one's speech sound natural, and even if using "recites" here is technically grammatical, that doesn't make it sound any less unnatural to native speakers (particularly when such an easy, more suitable alternative for a non-legal register exists in "read").
Best Answer
To put it simply, I would say that grasp means "beginning to understand" something difficult or obscure, as opposed to understand, which implies a rather complete and satisfactory comprehension of the matter.