For a fixed integer $n\ge2$, we can define an equivalence relation in the ring $(\mathbb Z,+,\cdot)$. That relation is known as "congruence modulo $n$", "equality modulo $n$", or "equivalence modulo $n$". It's noted $u\equiv v\pmod n$ when integers $u$ and $v$ are in said relation, meaning
- $u$ and $v$ are such that $v-u$ is some multiple of $n$
- that is, there exists an integer $k$ such that $v-u=k\,n$.
and said equivalence relation is used to build the residue class ring $\mathbb Z/nZ$, which (in crypto) is often noted $(\mathbb Z_n,+,\cdot)$ or just $\mathbb Z_n$.
$v\bmod n$, without parenthesis immediately on the left of$\bmod$ nor $\equiv$ sign at the same level of the expression, can be defined as the smallest non-negative member of the infinite set of integers $u$ with $u\equiv v\pmod n$. It holds $u=v\bmod n$, where$\bmod$ is an operator.
The notation $y=x^e\bmod n$ implies $0\le y<n$, but $y\equiv x^e\pmod n$ does not. Thus $y=x^e\bmod n$ defines a unique integer $y$ as a function of $x$, $e$ and $n$, when $y\equiv x^e\pmod n$ does not.
The output of the (textbook/raw) RSA encryption function $x\mapsto y=x^e\bmod n$ (with $x$ an integer and $0\le x<n$, which I assume in the all following) is a uniquely defined integer $y$, of size at most that of $n$. In particular (for $n>2$ and $e>1$) that function can't simply return $y=x^e$ for all $x$, which $y\equiv x^e\pmod n$ allows.
The difference matters because $x\mapsto y=x^e$ is a function that's easy to invert (by extracting an $n^\text{th}$ root in the integers); but with $n$ and $e$ properly chosen for RSA, the (mathematically invertible) function $x\mapsto y=x^e\bmod n$ is conjectured to be computationally hard to invert given $n$, $e$, and random $y$ or $y$ for random unknown $x$, unless one can obtain the factorization of $n$ or some equivalent information like $d$.
The modulo calculation is always very simple: $(s^e\,y)^d\equiv s^{e\,d}*x^{e\,d}\equiv(s\,x)^{e\,d}\pmod n$
Note: the modified equation clarifies that here$\bmod$ is not an operator, but a qualifier of the equivalence relation $\equiv$
This is a fact holding for $(n,e,d)$ as in RSA, and all integers $x$, $y$, $s$ with $y\equiv x^e\pmod n$. It does not allow to factor $n$, nor compute from $(n,e)$ a $d$ making $y\mapsto x=x^d\bmod n$ the inverse function of $x\mapsto y=x^e\bmod n$, or otherwise invert that function for random $y$ or $y$ for random unknown $x$.
Said fact would allow some manipulations if the textbook RSA function $x\mapsto y=x^e\bmod n$ was directly used to encrypt $x$, or if it's inverse function $y\mapsto x=y^d\bmod n$ was directly used to sign $y$. Common practice prevents such manipulation by choosing $x$ or $y$ close enough to random.
