The key step in Zermelo's proof of the well ordering theorem is to use $\text{AC}$ to simultaneously choose the next elelment for all possible partial chains in prospective well orderings, but that requires forming the set of all partial chains first, hence the powerset axiom $\text{P}$, so that the choosing can occur. And it is the well ordering of the continuum that leads to most "paradoxes" usually blamed on $\text{AC}$.
People objected to completing a process of picking elements from 'too many' sets to form a set, why not to completing a process of creating too many sets to form a set?
Now concerning the consequences. There are many books detailing which forms of $\text{AC}$ are needed for which theorems of analysis and algebra, but I could not find any detailed accounts for $\text{P}$. There is a bold claim that "we can do practically all mathematics without the axiom of powerset" here, but the references are not particularly illuminating, and in some sense this can be said even of Peano arithmetic.
This is an interesting paper, but it deals more with logical and set-theoretic consequences of dropping $\text{P}$ than with its effect on 'ordinary' mathematics. It turns out that collection doesn't follow from replacement without $\text{P}$, and using it makes for a better theory denoted $\text{ZFC}^-$. Then there is an obvious issue of not being able to form real numbers, but there are similar 'catastrophes' if at least countable forms of $\text{AC}$ are not assumed. So in fairness let's assume that collection of at most countable subsets of a set is a set (such a set is formed by running only simple inductive processes "in parallel", like real numbers from decimal expansions).
How much of classical mathematics survives in $\text{ZFC}^-+\text{P}_\omega$ or similar theory? Is it so little that it's not worth considering?
For the purpose of applying set theory as a foundational subject, one must be able to express typical mathematical reasoning, such as
And in general, if you have a kind of thing $T$, mathematicians tend to become interested in sets of things, functions whose domain or range consists of things, predicates on things, and so forth.
The usual application of set theory requires there to be a set encompassing all of the choices. We need a set of real numbers, a set of integers, a set of sequences of real numbers, a set of real-valued functions of the reals, a set of subsets of the reals, a set of sets of things, a set of functions whose domain or range consists of things, a set of predicates on things, and so forth.
And thus, you need power sets.
Note that if you used function sets rather than power sets as your basic idea, you'd still get something that adequately serves the purpose of power sets in the form of the set of functions $S \to \{ 0, 1 \}$.
If you give up power sets, you have to find a whole new approach.
Incidentally, while rejecting the axiom of powersets kills typical mathematical constructions, rejecting the axiom of choice does not: even without the axiom of choice, you can still form arbitrary infinite products of nonempty sets
$$ \prod_{i \in I} S_i $$
it's just that you no longer have a guarantee that this product is nonempty. You only kill reasoning that depends on the construction giving nontrivial results.