In response to Semyon's question in the comments: The concept of a $q$-analogue in combinatorics has never been entirely rigorous, and if anything the construction of quantum groups has been clarifying. The rough idea is that a counting problem in combinatorics is interesting when it has a "nice" answer, which often (but not by any means always) means an efficient product formula. So then a $q$-analogue is a weighted enumeration or finite generating function in which every weight is a power of $q$, and the enumeration still has all favorable numerical properties, and $q$-integers or cyclotomic factors arise.
In the case of quantum groups, first of all they are Hopf algebras. A Hopf algebra is an algebra together with all necessary extra apparatus to define the tensor product of two representations as a representation (i.e., comultiplication) and the dual of a representation as a representation (i.e., the antipode map). A universal enveloping algebra $U(\mathfrak{g})$ is of course a Hopf algebra. In this case, any deformation of $U(\mathfrak{g})$ as a Hopf algebra is potentially interesting. There is a cohomology result that if $\mathfrak{g}$ is complex and simplex, then there is only one non-trivial deformation, and you might as well call its parameter $q$ with $q=1$ at the undeformed point. (Sometimes the logarithm of $q$ is used and is called $h$, in reference to Planck's constant.) Since this is the only deformation, it is an analogue of some kind, and it is interesting. Moreover, there is a parametrization of the deformation so that $q$-integers (or quantum integers, in centered form) and cyclotomic polynomials arise in the structure of the Hopf algebra and its representations. The analogue thus deserves to be called a $q$-analogue.
Theo's explanation illustrates this more explicitly. The quantum plane is a non-commutative algebraic space (in the sense that you can interpret it as a purely formal "Spec" of the quantum plane ring) that is associated to the non-cocommutative algebraic group version of $U_q(\text{sl}(2))$. So, then, in the ring of the quantum plane, if you just expand $(x+y)^n$, you get a $q$-analogue of the binomial coefficient theorem using Gaussian binomial coefficients. (Where the $q$-exponent of a word in $x$ and $y$ is its inversion number, just as with the $q$-enumeration of permutations.) This is one of many examples where $q$-analogues that were considered long before quantum groups appear in the theory of quantum groups.
As for knots and links: In order for $U(\mathfrak{g})$ to have a non-trivial deformation as a Hopf algebra, you have to allow comultiplication to be non-commutative, even though $U(\mathfrak{g})$ itself is cocommutative. So then if $V$ and $W$ are two representations, $V \otimes W$ and $W \otimes V$ are not isomorphic via the usual switching map $v \otimes w \mapsto w \otimes v$, because that switching map is not in the category. (Due to non-cocommutativity, it is not equivariant, i.e., not an intertwiner.) However, $V \otimes W$ and $W \otimes V$ are still isomorphic, just by an adjusted version of the switching map. (You see the same theme in the category of super vector spaces, where there is a sign correction when $v$ and $w$ both have odd degree.) However, there are two natural, in-category deformations of the switching map, not just one. It so happens that they should be interpreted as left- and right- half twists in a braid group, so that you get braid representations and ultimately knot and link invariants.
The point is that ordinary tensors (and tensor networks) live in spaces that have natural actions of symmetric groups, because you can permute indices of tensors. The whole theme of quantum group definitions is deformations, and it so happens that the symmetric group action deforms into a braid group action.
This explains topological invariants such as the Jones polynomial in a one- and two-dimensional sort of way, from braids to diagrams of knots to knots themselves. It is more satisfying to have a more intrinsically 3-dimensional definition. (Actually, what counts as intrinsically 3-dimensional is somewhat debatable, but never mind that.) This is why Witten provided a "definition" of the Jones polynomial and related invariants using Chern-Simons quantum field theory. It is not really a rigorous definition, but it is very credible as a "physics definition" or even an incomplete, but maybe one-day rigorous, mathematical definition. This leads to the basic association between Chern-Simons quantum field theory and quantum groups, that they are two ways to describe the same topological invariants.