Analogy between "One and only one" and "If and only if"
To show the existence of an X satisfying Y it is sufficient to prove "existence" of such an X: that there is "some" (one or more) X satisfying Y.
Then, to show that there is only one such X, you need to show that if K also satisfies Y, then K must equal X.
So "one and only one" requires establishing both (1) and (2): existence and uniqueness.
Put differently, "there is one and only one" can be read as the conjunction of:
(a) existence of "at least one" X such that X satisfies Y,
... and ...
(b) existence of at most one such X that satisfies Y.
If we let $P(x)$ denote the satisfaction of some property $P$ by $x$, then we can assert that there is one and only one $x$ such that $P(x)$ as follows:
$$\exists x[P(x) \land \forall y(P(y) \rightarrow y = x)]$$
For the difference between: (i) "only one X satisfies condition Y" and (ii) "there is one and only one X satisfying Y": One can argue that there might be some property that only one element could possibly satisfy, without necessarily asserting that therefore, such an element exists (ii).
Using "there is one and only one" or "there exists a unique" (or even "there exists exactly one") is less ambiguous than stating "only one", which can be taken to mean "at most one."
"If and only if" is a two-way implication, i.e. the left implies the right and the right implies the left. "One and only one" is a statement regarding existence and uniqueness. There is not necessarily a two-way relation here. For instance, the solution to a quadratic can exist, but need not be unique.
They are related because the statement "There is one and only one $x\in U$ such that $P(x)$" is equivalent to "There is $x\in U$ such that $P(y)$ if and only if $y=x$.