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:
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:
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.