Difference between $\mathbb{Z}/n\mathbb{Z}$ and $\mathbb{Z}_n$ In every Modern Algebra book I've read, I've seen that the groups $\mathbb{Z}/n\mathbb{Z}$ and $\mathbb{Z}_n$ are isomorphic, but not equal. I understand the difference between "isomorphic" and "equal," but this particular example raises a couple of questions for me.

I know the first group consists of cosets $k + n\mathbb{Z}$, and the second group consists of equivalence classes $[k]_n$, but isn't it true that $k + n\mathbb{Z} = [k]_n$ for every integer $k$? (They are both sets containing the same elements from $\mathbb{Z}$.) And if so, then can't we say that $\mathbb{Z}/n\mathbb{Z} = \mathbb{Z}_n$? Thanks in advance!
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.
No , I am suggesting that Z_n is the group multiplicative group of nth roots of unity as I stated above . Surely the books you referred to give a definition for Z_n . How can they mention a group without telling you what it's elements are .If not take my definition ,it's right .Also it is important because is gives a natural connection between the multiplication and addition operations . You can do the same with the additive group of R (real numbers ) and the multiplicative group of the Positive Real numbers via x ---> e^x . Here no quotient group is involved  and they are infinite groups .  Again in your example Z/nZ and Z_n are specific groups with definitions given or known . Of course all this assumes that C (the field of complex numbers and R (real numbers) are fixed throughout ,Of course they have many different constructions better forgotten.(sometimes )