Once in a while when a question like "how I get good random numbers" is asked the suggested approach is to just generate an UUID. UUID looks like a random number and it is designed in such way that UUIDs likely never repeat - it is unique.
What is mathematical difference between "unique" and "random"? Why exactly should UUID not be used as a random number - for example as a cryptographic key?
When you say random numbers you want to have a series of numbers such that given any sequence of $i-1$ numbers, the probability of the $i$th number having a certain value is independent from all its predecessors (I (over)simplify here), i.e.
$\begin{equation*}Pr[n_i = k \mid n_0, \dots, n_{i-1}] = Pr[n_i = k]\end{equation*}$
Unique numbers can be generated without having to guarantee that (e.g. just counting upwards yields unique numbers). These kinds of unique numbers can be guessed, which make them bad random numbers.
In fact, most things that are called random-number generators in practice are pseudo-random-number generators. That means that they have a perfectly deterministic and predictable algorithm of constructing the next number but they start with a secret seed that changes from application to application. Without knowing that seed the generated series have many properties of true random sequences. That can be checked by certain statistical methods.
In most practical, every-day scenarios you want to have unguessable sequences, e.g. for password creation. Pseudo-random-number generators are fine for that provided you choose a proper seed; many people use the system timestamp when the request comes in, assuming this is pretty hard to predict up to the millisecond. If you really need the statistical properties of random sequences, for example for Monte Carlo simulations, you should be more wary.
Edit: To clarify: Random sequences (R) and unique sequences (U) are not opposing concepts; you can have all combinations of both:
I guess the main source of confusion is that there are many versions (and many implementations) of UUIDs.
Random but not unique: As you can see from the Wikipedia article, version 4 UUIDs are just 122-bit random numbers. Nothing else. They are exactly as good for cryptographic purposes as the random number generator that you used to implement it. And they are not "unique" in any other sense except that it's highly unlikely to get collisions (at least if you use a good random-number generator).
Unique but not random: In the other extreme we have version 1 UUIDs, which concatenate MAC addresses with timestamps. Assuming that the MAC addresses are globally unique (as they are supposed to be, as there is a central authority that allocates them), and assuming that you don't generate UUIDs too rapidly, then the end result is something that is guaranteed to be globally unique. However, it is not "random" in any sense; if you know who generated the UUID and when, you can predict the value of the UUID fairly accurately. Obviously, it is an extremely bad idea to use such UUIDs as a replacement of random numbers in any cryptographic application.
An object is mathematically unique, if there does not exist another object which satisfies the properties of the first object. For instance, the solution of $7x+2=5$ is unique (over reals), while the solution of $x^2=4$ is not unique (over reals).
An object is practically unique, if there is not an efficient procedure which finds another object that satisfies the properties of the first object. For instance, let $H(\cdot)$ be a cryptographically secure hash function. By definition, the solution to $y=H(x)$ is (practically) unique; That is, while there are mathematically infinitely many $x$'s which satisfy $y=H(x)$, there is no efficient procedure which, given $y$ and $x$, can find $x' \ne x$ such that $y=H(x')$.
This is the case with UUID's. Given a (random) seed, the UUID generator produces a (practically) unique UUID; That is, it is practically infeasible to find two seeds $s \ne s'$ such that $UUID(s) = UUID(s')$, while mathematically such seeds exist.
When I teach this concept in the class, I usually provide an example: Assume that, to find such a collision, one must try $2^{128}$ seeds. This number is so high that it is beyond the (estimated) amount of particles in the known universe!
