What is the significance of nominal techniques?
Short answer. Formal reasoning about binding and $\alpha$-conversion with  nominal approaches is closer to intuitive reasoning than alternative approaches.

Longer answer. Binders arise everywhere in mathematics and computer science. 
Dealing with binders has historically been done in a handwaving way (e.g. assuming Barendregt's variable convention). This works well in informal mathematics, but breaks down in formal reasoning. In implementations of programming languages, this problem was initially handled with some form of renaming using a global operation, often called gensym, that creates globally fresh variables. Already in the implementation of programming languages that has some disadvantages, but the problem comes to the fore when reasoning about formal languages with binders, in particular about certified software. 

The problem is that in early formal proofs about languages with binders, a disproportional amount of effort was dedicated to proving largely uninteresting lemmas about substitutions and $\alpha$-conversion. The core of the problem is that programming language  syntax is usually defined inductively. For example

$$
  P \quad ::= \quad x = E \ |\  while\ P\ do\ P \ |\   ...
$$

Given this genesis of programming language
syntax, it is natural to prove program properties by structural induction on program syntax. This works fine for simple languages like that above, but once we add binders

$$
  P \ ::=\ ... \ |\ \lambda x.P
$$

the situation changes. $P$ is not a subterm of $\lambda x.P$. We want to identfy programs up to consistent
renaming of bound variables, for example we don't want to distinguish
$\lambda x.ax$ from $\lambda y.ay$. In other words, programs are
equivalence classes ... and structural induction goes out of the
window. 

Several techniques have been proposed for a better mathematisation of  the informal handling of binding.


Higher-order abstract syntax,
De Bruijn representation of binding, 
Getting rid of variables altogether by using combinators.
Nominal technique which replaces substitution by swapping. Swapping is essentially* a first-order operation and can be internalised in the ambient logic.


Of all those proposals, the reasoning about binders with nominal techniques is closest to the intuitive "handwaving" done to handle binders in informal reasoning. That is the key advantage of nominal techniques. This might be best observed in Nominal Isabelle, where most of the reasoning about binders can be hidden from the user. See Nominal Techniques in Isabelle/HOL for a discussion. 



* There is a subtle issue  in that 
if you want to define higher-order functions by recursion in nominal logic, the recursion operator requires some kind   of freshness. See the conclusion of Pitt's Nominal logic, a first order theory of names and binding, which points out that "Nominal Logic does not give a complete axiomatisation of the
notion of finite support that underlies the notion of freshness in FM-sets. Nevertheless, the first-order
properties of a notion of freshness of names presented in this paper seem sufficient to develop a useful
theory, independent of any particular object-level language involving binders".