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The terms classifier and measure word are often used interchangeably, with the latter being (reportedly) more common in language courses.
They both translate to 量词 (literally "measure word") and their grammatical function is essentially the same.
So is there any actual difference between classifiers and measure words in Mandarin, or do the two terms identify the exact same thing?
Yes, there is a slight difference (although it appears there's no complete consensus on this).
A classifier, in linguistics (not just in Mandarin), is a word or morpheme used to express that a countable noun belongs to a semantic class. Here "semantic class" means a category of referents with some common traits; if we think about how classifiers work in Chinese, this is easy to understand.
For example, the classifier 条 is used for long, narrow and flexible objects. So when we say:
一条鱼
we are implying that the fish belongs to a class of things that, more or less, share the same properties: long, narrow and flexible. That is the semantic class, and that's why the classifier is called classifier.
Therefore, by using a different classifier, we offer a different classification of that object. When we say:
一群鱼
we are conveying that fish belongs to the category, or class, of things that go in "groups" (in "schools" actually!!!).
Now, English does have measure words: these are mostly nouns with their own meaning that work as units of measurement for mass nouns, which are not countable. They come in the form a something of <noun>. Let's consider the word water. In English we can't count it ("one water", "two waters"). So when we need to quantify a certain amount of water, we have to use a word that describes a precise unit of measurement, for example:
a drop of water
a liter of water
an ocean of water
This definition of measure words works in Chinese too. The difference being that in Chinese there are classifiers proper and measure words. So what's actually the difference?
Classifiers are not standalone words, they don't have meaning on their own; and as in English, they still can't accompany mass nouns (consider 两个水 vs 两升水)
Measure words are words with a precise meaning, which can also work as units of measurement; and they can accompany mass nouns.
Her and Hsieh, On the Semantic Distinction between Classifiers and Measure Words in Chinese propose also the following definitions, based on tests to appraise whether a certain counter word is a classifier or a measure word:
Classifiers classify or categorize nouns by highlighting some salient or inherent properties of the noun and thus contribute no additional meaning.
Measure words, on the other hand, play a substantive role in denoting the quantity of the entity named by the noun.
So, saying 一条鱼 does not add new information to what we already know about a fish. A fish is inherently a long and flexible object (mostly; an inflated puffer fish maybe is not "long"). Hence, 条 is a classifier proper. Whereas saying 一箱鱼 does add new information, we are saying that the fish comes in crates, where fish, in itself, doesn't inherently come in crates. Hence, in this example 箱 is a measure word.
In conclusion, classifiers proper stress the existence of an essential feature of the counted noun, whereas measure words stress a transient, not inherent (accidental) numerical feature.
Nouns denote things, physical or abstract. Based on the primitive properties of the things (this perception may vary according to socio-cultural background), we distinguish between "discrete nouns" and "non-discrete nouns".
Nouns denoting physical things are usually discrete nouns. For example: an apple, a book, a tree, an aeroplane..., etc. These things come with discrete shapes or forms.
Nouns denoting abstract things are compact nouns (at the other extreme, as opposed to being discrete). For example: health, peace, poetry, philosophy, biology..., etc. These things have no discrete froms.
In between, there is another category of non-discrete nouns which we can call "dense nouns". For example, water, wine, sand, hair..., etc. By "in between", we mean that, like compact nouns, these things have no discrete forms, but can be "separated" and "measured" using another thing which has a form. For example, "a glass of water", "a bottle of wine", "a spade of sand". This "other thing" serves as kind of measure. As such, it can be called a "measure word".
To indicate a discrete object in English, we use "a + thing". In Chinese, we use "classifier + thing". The classifier is related to the discrete form of the object: 条 for linear objects, 块 for flat and hard objects, 粒 for small and roundish objects..., etc. It depends very much on people's perceptual habits and sometimes can be quite arbitrary. Indeed, classifiers vary from dialects to dialects.
When we use a non-discrete noun with a measurable quantity in Chinese, we can use 些 or 点: 他喝了些酒，我买了点橄榄油. In this case, 些/点 acts like a measure word, only that it does not have a form. It just indicates a "measurable quantity".
When we use 些/点 with a discrete noun, the idea of "measurable quantity" will override the idea of "discreteness": 我刚吃了点水果，他们养了些鸡和鸭.
Therefore:
May be due to my Cantonese background I have a different understanding of the difference between 'classifier' and 'measure word as Chinese grammar terms.
For me, classifier classifies objects, base on their shape, size, substance, grouping, and so on. e.g. 一條魚，一粒米， 一灘水， 一盒餅
A classifier itself is not a measure word until it is used with a counting word.
In other words, "measure word' is another name for [quantifier/ counting word + classifier], e.g. 幾條魚，兩粒米
~
Cantonese Example of classifier used as the determiner 'this':
條魚好新鮮 - this (stick of) fish is very fresh
邊條好新鮮? - which stick (of fish) is very fresh?
嚿肉丸好硬 - this (piece of) meatball is very hard
呢嚿就好軟 - this piece (of meatball) is very soft
To me, classifiers belongs to measure words. In other words, classifiers is a type of measure words. The idea of classifiers offers a way to construct a type in order to count that type of things. The essential is just for measuring.
That could be the reason why we don't usually distinguish the two terms in practice. And in Chinese, we'd mark it as 量词 all the time and no one cares whether it would fall into its subset 分类词 because it doesn't affect any semantics so they don't want to waste their effort to further trace down technically.
I'm not sure if you should distinguish the difference between classifiers and measure words in English. That seems to be an English question.
My two cents.
All the other answers were very insightful. I also find appropriate to reference the instructive Wikipedia entry for Chinese classifier:
The terms classifier and measure word are frequently used interchangeably (as equivalent to the Chinese term 量词 (量詞) liàngcí, which literally means "measure word"). Sometimes, however, the two are distinguished, with classifier denoting a particle without any particular meaning of its own and measure word denoting a word for a particular quantity or measurement of something, such as "drop", "cupful", or "liter". The latter type also includes certain words denoting lengths of time, units of currency, etc. These two types are alternatively called count-classifier and mass-classifier, since the first type can only meaningfully be used with count nouns [nouns organized in discrete units, such as apples], while the second is used particularly with mass nouns [nouns organized in continuous units, such as water or rice].
The difference between count-classifiers and mass-classifiers can be described as one of quantifying versus categorizing: in other words, mass-classifiers create a unit by which to measure something (i.e. boxes, groups, chunks, pieces, etc.), whereas count-classifiers simply name an existing item. Most words can appear with both count-classifiers and mass-classifiers; for example, pizza can be described as both 一张比萨 (yì zhāng bǐsà, "one pizza", literally "one pie of pizza"), using a count-classifier, and as 一块比萨 (yí kuài bǐsà, "one piece of pizza"), using a mass-classifier. In addition to these semantic differences, there are differences in the grammatical behaviors of count-classifiers and mass-classifiers; for example, mass-classifiers may be modified by a small set of adjectives (as in 一大群人 yí dà qún rén, "a big crowd of people"), whereas count-classifiers usually may not (for example, *一大个人 yí dà ge rén is never said for "a big person"; instead the adjective must modify the noun: 一个大人 yí ge dà rén). Another difference is that count-classifiers may often be replaced by a "general" classifier 个 (個), gè with no apparent change in meaning, whereas mass-classifiers may not. Syntacticians Lisa Cheng and Rint Sybesma propose that count-classifiers and mass-classifiers have different underlying syntactic structures, with count-classifiers forming "classifier phrases", and mass-classifiers being a sort of relative clause that only looks like a classifier phrase. The distinction between count-classifiers and mass-classifiers is often unclear, however, and other linguists have suggested that count-classifiers and mass-classifiers may not be fundamentally different. They posit that "count-classifier" and "mass-classifier" are the extremes of a continuum, with most classifiers falling somewhere in between.
Researchers have differing views as to how count-classifier – noun pairings arise: some regard them as being based on innate semantic features of the noun (for example, all nouns denoting "long" objects take a certain classifier because of their inherent longness), while others see them as motivated more by analogy to prototypical pairings (for example, "dictionary" comes to take the same classifier as the more common word "book"). The use of classifiers did not become a mandatory part of Old Chinese grammar until around 1100 CE. Some nouns became associated with specific classifiers earlier than others; the earliest probably being nouns that signified culturally valued items such as horses and poems. Many words that are classifiers today started out as full nouns; in some cases their meanings have been gradually bleached away so that they are now used only as classifiers.
I also found curious the case of words which are their own classifiers (or count-classifier), such as 天 (day) or 年 (year). Thus we say:
一天
one day
Instead, these still accept an additional measure word (or mass-classifier):
一些天
a few days
