Difference between $\mathbb{Z}/n\mathbb{Z}$ and $\mathbb{Z}_n$

Suppose we have a surjective homomorphism $h: G \to H$. The distinction, in my mind, between $\Bbb Z/n\Bbb Z$ and $\Bbb Z_n$ is the same as the distinction between $G/\text{ker }h$ and $H$.

Clearly, these are isomorphic groups (they have the same algebraic properties), but they are distinct in their "set structure" (the quotient group elements are elements of the power set of $G$, and the homomorphic image elements are just set elements (singletons) (of $H$)). In some areas of math (cough, topology) it can be crucial to keep track of "which level of set-construction" you are in.

Here is a similar example (from vector spaces) to illustrate the "ontological" versus "abstract" distinction.

The $x$-axis is a (normal) subgroup of the additive (vector addition) group of the Euclidean plane. We can form the coset space of all horizontal lines, which is isomorphic to, but surely not equal to, the $y$-axis.

As algebraic objects, we "abstract away" the particulars of how a given group arises, as we are usually only interested in its properties. However, the group may arise in a particular context we are studying, where its basis of definition may matter. One should not, for example, confuse the congruence class modulo $n$ of the integer $k$, with the integer $k$ itself.

This is the nature of algebraic morhpsims in general-they "filter away" information. In some sense, the quotient object has "more information to keep track of" (in $\Bbb Z/5\Bbb Z$, for example, one has to perform a (simple) mental calculation to resolve $7 + 5\Bbb Z$ to $2 + 5\Bbb Z$), whereas the homomorphic image has discarded the excess baggage.

Let's consider another example, and see if we can tease out what we want from our definition of equality and isomorphic.

One group is the rotations and reflections of a square which leave it in the same place it started (though the points inside may be moved about). The other group is generated by $2$ elements $a,b$, and follows the relations: $a^4 = e$, $b^2 = e$, and $ab = ba^3$. It turns out these two groups are isomorphic - we can identify $b$ with horizontal reflection, and $a$ with rotation by $\frac{\pi} 2$.. $b$ with vertical reflection and a with $-\frac{\pi} 2$. Further, if I draw a heart inside the square, I know what it means to reflect or rotate that heart. I can immediately apply the group operators for the first group. But what does it mean to apply $a$ to that heart?

Let's also look at a case where we can agree $2$ objects are equal. For example, $1 + 1 = 2$. The first sentence would have made just as much sense had I written "we can agree 1+1 objects are equal." (though linguistically horrific). The two representations, $2$ vs $1 + 1$, are interchangeable everywhere, and there's no new ambiguity from using one or the other. By this, I'd suspect that one would not say that the two groups I mentioned above are equal, but everyone would agree they're isomorphic.

Now, to return to the question for $\mathbb{Z}_n$ and $\mathbb{Z}/n\mathbb{Z}$. We agree they're certainly isomorphic, now we want to decide if they're equal. But first, a note on definitions. here and here both use $\mathbb{Z}_n$ and $\mathbb{Z}/n\mathbb{Z}$ for two different algebraic objects! Worse yet, they both note a third meaning for the notation $\mathbb{Z}_n$: the p-adics. Hence, to decide if they are different, we first need to agree on a definition for both. If we use yours, the first two of these links seem to suggest they're equal - but I can easily imagine a text defining the finite cyclic groups as $\mathbb{Z}_n$ and the equivalence classes as $\mathbb{Z}/n\mathbb{Z}$. Now, some sentances to consider (with ones definition of choice for $\mathbb{Z}_n$):

$[0]_5$ is a principle ideal. VS $0 + 5\mathbb{Z}$ is a principle ideal. (similarly for 'additive subgroup of $\mathbb{Z}$', or 'maximal ideal', etc)

$[k]_n$ is an equivalence class. VS $k + n\mathbb{Z}$ is an equivalence class.

$|k + n \mathbb{Z}|$ is infinite. VS $|[k]_n|$ is infinite.

$7$ is a member of $k + n \mathbb{Z}$. VS $7$ is a member of $[k]_n$

A few of these make me a bit nervous/uncomfortable - there seems like there could be some ambiguity between sending $[k]_n$ to $k + n \mathbb{Z}$ vs $-k + n \mathbb{Z}$ - as additive groups, these both look fine.

To answer this more concretely, one would need to know who asserted the two symbols were different, and their definitions of the two groups (as opposed to the most memorable definitions you've found).