There is no difference in terms of meaning.
Both words 「メモ」 and "memo" in respective languages have the primary meaning of a noun for "short note" (in general). The other meaning "memorandum" (in business) however, is not applicable to Japanese and only applicable to English counterpart (Details at below).
Direct answers
What differences, if any, are there between the English word "memo" and the Japanese word "メモ"?
No difference in terms of meaning.
In terms of usage however, the Japanese word counterpart could do verbing. So the sentence 「メモを取る」 can be expressed by a single word 「メモる」 to mean "to take note", albeit informal. You can look up in Japanese dictionaries (online, might not be found in printed).
Now try verbing with the word "memo" in English. Are you "memoing"? Even this entry on Wiktionary suggests that possibility, I can't seem to find the word "memoing" in English dictionaries to this answered date.
Does the Japanese word "メモ" have connotations to do with memorisation or memory that don't exist with the English word "memo"?
No, it doesn't.
Also, do native speakers of Japanese associate the word "メモ" with the English word "memory"?
No. At least I have never heard of any native Japanese using the word in such way. Japanese use either 「メモリ」 or 「メモリー」 to translate the English word "memory".
Finding related to "memory"
If the word 「メモ」 has anything to do with "memory", there is a suffix-forming noun in Japanese 「メモ化」 that is translated as "memoization" according to this entry on Wiktionary and this article on ウィキペディア.
The latter is linked to this article on Wikipedia, which briefly explains the origin of word and confusion between "memo-i-zation" and "memo-ri-zation".
The term "memoization" was coined by Donald Michie in 1968^ and is derived from the Latin word "memorandum" ("to be remembered"), usually truncated as "memo" in American English, and thus carries the meaning of "turning [the results of] a function into something to be remembered." While "memoization" might be confused with "memorization" (because they are etymological cognates), "memoization" has a specialized meaning in computing.
^ Michie, Donald, "Memo Functions and Machine Learning," Nature, No. 218, pp. 19–22, 1968.
Above all, there is slim chances of OP had misheard or misread the term "memorization" (for remembering) with "memoization" (in computing). The latter could have lead OP to the word 「メモ化」 which does not carry the same meaning as 「メモ」 despite sharing the same root word.
Updated 2017.09.13
Apparently, Japanese does not use 「メモ」 but 「覚書{おぼえがき}」 to mean "memorandum". This has been noted in this Japanese article on ウィキペディア. Although the former may be seen when explaining the actual word entry for "memorandum" i.e. this definition on コトバンク, these words 「メモ」 and 「覚書」 do not seem to be interchangeable.