Why is uniqueness important for PDEs?
Why do you want to solve the PDE in the first place? Usually because it relates to the solution to a real world problem.
Say, for example, that you're studying the temperature in a heated metal. You have to understand if the temperature will stay below a certain value, otherwise the metal will collapse and your bridge with it. You find out that the temperature solves a PDE with certain boundary conditions, you solve it and find a value for the maximum temperature.
Now, if the solution to the PDE is unique, then you are done. But if the solution is not unique, then who's to say that the solution you found is what will actually happen in the real world?
In the real world there is only one solution! You then have to specify additional boundary conditions until you're left with a unique solution. To do all this you need some theory that tells you exactly when your boundary conditions are enough, what can go wrong, where you should look for exceptions, etc.
In general this is important not only for real world applications; uniqueness hugely simplifies problems. Say that you have to solve for your abstract math problem a complex PDE; if you can "guess" the solution, and it works, then it works! You're done, that is the only solution and you go forward. If there were multiple solutions you would probably need to check if the solution you found is "the right one" for the problem at hand. It is useful in all sorts of context. Just think how useful is the fact that we know that a polynomial of degree $n$ has exactly $n$ complex roots (counted with their multiplicity). It makes life much easier in a whole lot of ways
Let's think practically.
PDE's is a field used primarily for physical modelling. Almost all introductory texts talk all about physics models using PDE's: Heat equation, transport equation, wave equation, etc.
In a real life scenario you see a behaviour and you want to model it by using math. For example, you heat up an insulated rod and you want to predict the temperature at any time using math. If you did not have uniqueness we would have two possible end results. For instance, at time $t=3$ seconds we could have that the temperature of the rod is $32^o C$ or $-76^o C$ based on the results of two solutions. This is of course physically ridiculous. This means that the model we have proposed does not replicate the reality of what is happening. So we must pose a new model with a unique solution as we only expect one result in testing.
For P/LDEs, recurrences, etc. uniqueness theorems are powerful tools for proving equalities. Let's consider some very simple examples.
$e^{ix}$ and $\, \cos(x) + i\,{\rm sin}(x)\,$ are solutions of $\ y'\! = i y,\ y(0) = 1\,$ so they are equal by uniqueness.
$\sum_{k=1}^n k\cdot k! = (n\!+\!1)!-1\,$ since both satisfy $\,f(n\!+\!1)-f(n) = (n\!+\!1)(n\!+\!1)!,\ f(0) = 0.\,$ Here the uniqueness proof for the first-order difference equation (recurrence) has a trivial one-line inductive proof (sometime called the Fundmaental Theorem of Difference Calculus). See also here for a vivid $2$-dimensional version using differences of rectangles.
$\prod_{i=1}^{n-1}\left(1+\frac{1}{i} \right)^{i} = \frac{n^{n}}{n!}\ $ since both satisfy $\,f(n\!+\!1) = (1+1/n)^n f(n),\,\ f(1) = 1.$
Binet's formula for fibonacci numbers $\, {f}_n = (\phi^n-\bar\phi^n)/(\phi-\bar\phi),\ \phi = (1+\sqrt 5)/2\,$ is very easily proved by showing the RHS satisfies $\,f_{n+2} = f_{n+1}+f_n,\,\ f_0 = 0,\ f_1 = 1.$
Combining such uniqueness theorems with the group-theoretic machine for special functions, some Lie theory (symmetry and separation of variables), and work on holonomic functions, one obtains very powerful algorithms for effectively computing with special functions. These lie at the core of many computer algebra systems such as Macsyma, Maple, Mathematica.