What is the significance of Friedlander-Iwaniec and related theorems?

I think questions about prime values of polynomials are considered inherently interesting.

All these questions are special cases of Bunyakovsky's conjecture, or, if you want, the Bateman-Horn conjecture. Certainly mathematics is not yet ripe for such problems.

You can also view them as special cases of the general problem of when special sequences of numbers are prime. One generally expects to see transfers of techniques from one special case of this problem to another, but not as much with other areas of number theory and mathematics. So to a large extent I think the results on primes-of-a-special-type problems that are considered most interesting are those that point to way to further solution of primes-of-a-special-type problems.

One could also note that primes of the form $a^2+b^4$ is a special case (in some sense) of primes of the form $a^2 +1$, since infinitely many primes of the form $a^2+1$ implies infinitely many primes of the form $a^2+b^4$. (Of course this is the reverse of the special case direction between the numbers themselves.)

I don't think many who work on these problems are motivated by applications to cryptography. As Ben Smith points out in the comments, there are some applications of Bateman-Horn to cryptography, where we would like to find primes represented by special polynomials for use in certain cryptographic algorithms. Conditional on not just the existence of such primes but the Bateman-Horn conjecture predicting how many there are in a large interval, we can generate such primes in polynomial time by randomly sampling values of the polynomial and primality testing.

However, since these conjectures are very very hard to prove, but we have very strong reasons to believe that they are true, working on them might not help with cryptography very much.

Suppose you suspect that among some subset $\mathcal{S}$ of the natural numbers, there ought to be infinitely many primes. How would you confirm that this is indeed the case?

Clearly, the "bigger" $\mathcal{S}$ is (measured in terms of natural density), the easier the question should be. The biggest set is of course $\mathbb{N}$ itself, and it was a highly non-trivial theorem of Euclid that confirmed this. What about the next most obvious "big" sets, the arithmetic progressions? Indeed, consider the set $\mathcal{S}(a; q)$ by

$$\displaystyle \mathcal{S}(a; q) = \{qx + a : x \in \mathbb{N}\}.$$

In the special case $q = 4, a = 3$ one can use Euclid's argument again to show that there are infinitely many primes in the set (but far from the correct density). A much more difficult argument shows that the case $q = 4, a = 1$ also gives infinitely many primes. But we expect that there are infinitely many primes in $\mathcal{S}(a; q)$ whenever $\gcd(a,q) = 1$. This is not known until the work of Dirichlet, almost 2000 years after Euclid!

The most extensive generalization of the theorem of Dirichlet known is the Chebotarev density theorem, which shows that in any number field there are infinitely many primes which split completely (over simplifying here).

So far, all of the sets considered have log density one. If we put $\mathcal{S}(X) = \# \{n \leq X : n \in \mathcal{S} \}$ then the log density is the infimum among all non-negative real numbers $\delta$ such that $\mathcal{S}(X) < X^{\delta}$ for all $X$ sufficiently large. Log density one sets can still be very very hard to count primes in: indeed, we expect the set of primes $p$ such that $p + 2$ is also prime to be infinite, and this set (if infinite) is expected to have log density one.

Thus, producing an infinite set of primes which have log density less than one is a formidable task. Both the Friedlander-Iwaniec theorem and Heath-Brown's theorem are examples where this can be achieved. These should be considered highly exceptional in the sense that similar, perhaps even easier looking polynomials are not amenable to their methods! Perhaps the most infamous is the polynomial $F(x,y) = 4x^3 - y^2$, which counts discriminants of elliptic curves in short Weierstrass form.