"I got this wallet from my mother."
I feel like I might be answering my own question here, but as others have asked, it looks as though there's an inconsistency with how Duolingo handles this sentence structure. Another question in this section asks to translate "I get shoes from my dad," which would become 父につくをもらいます.
This sentence structure above is wrong when put in this setting. Duolingo doesn't accept 母にこのさいふをもらいました. What exactly is the difference here? I have an inkling (hence the answering my own question part), that this question has the topic highlighted (wallet), which is why a topic marker is used (は). The shoes example is more of a general statement, which is why there is no topic marker.
I guess what my real question is if the distinction really matters, as they technically mean the same thing.
One major rule in Japanese is that placing something earlier in the sentence automatically adds equivalent emphasis on it. In this case, with the use of この, you denote the sentence being about this wallet specifically, you are on the right track. It's a matter of nuance and what is more in line with the natural rules of the language, but essentially it's the same difference between 'The ball was kicked by mom.' and 'Mom kicked the ball.'
母 (はは）is the word 'my mother', when speaking to someone outside the family (or close friends) about her. Neither お nor さん should be affixed to this form. はは should be considered a 'clinical' usage of the word 'mother', therefore politeness modifiers would be awkward and unnecessary.
お母さん (おかあさん）is more flexible. It can be used for direct address TO 'one's own mother' as well as address OF the mother of a second or third party (your/his/her/their mother).
母さん (かあさん): The お can be dropped when addressing one's own mother or referencing her during dialog within a family (or familiar) unit. In some households, おかん (which is a contraction of お母さん) is even more familiar/casual.
Motherhood as a concept can use 母 (はは), お母さん, ママ, etc. 'She's going to be a mother' = 「(彼女は）母/お母さん/ママ になる。」
So, while お母さん can be used somewhat flexibly, 母 (はは) cannot. One can infer from the usage of 母 (はは) that the speaker is talking to someone outside of the family & friend zone.
Both はは and かあ are 訓読み (Kun'yomi - Japanese native readings). There are other, less used, readings which need not be addressed now. The main 音読み (On'yomi - Chinese-derived reading) is ぼ.
お母 does exist as a word, but it would be pronounced おかか, which is 'baby talk' for 'mommy'.
So, the short answer is: 母 (はは) by itself and words which use 母 as an element should be considered as different words because their usage is different (despite referencing the same subject/concept).
I also have anecdotal evidence of the other form being used often (usually in more formal settings). Both are valid, still taught, and still used (and not only by old-fashioned people).
While many Japanese are unaware of the difference in nuance, に emphasizes the 'receipt' of a thing while から emphasizes who it was received from.
See this link (only in Japanese) for a QA about this topic.
More information on くれる、あげる、もらう at Tae Kim's page.
Checking contemporary usage statistics, から is used slightly more than に, but not by much.
I mean it just seems weird to place an honorific before a word for a common, everyday object, such as a wallet.
くれる has some very odd rules about it, it's only used when you yourself are specifically mentioned in the sentence (私、僕、俺 etc.). Otherwise when the object is mentioned as topic or whichever contextual reference to you is made instead, it's もらいます for receiving, あげます for giving (with に denoting whom the action is directed towards or from respectively).
You don't have to put から. I don't know where you're getting this from.
Using に emphasizes what what received, while から emphasizes who it was received from, but most Japanese use them interchangeably and only understand the nuance on a subconscious level. に is pretty much the default as far as もらう is concerned.
Here's just one reference.