We keep the geometric structure on tilings mainly because the tilings are generated with that structure, often from lattices or geometric group actions.

It's quite nontrivial that if you take some types of nice tilings, and forget the geometric structure, then you can indeed recover important information about the tiling from the graph alone. You can recover whether the tiling is periodic by looking at the group of symmetries of the graph. You can also recover information about the space, not just the tiling. You can recover whether the tiling is of the Euclidean or hyperbolic plane by looking at the growth rate of the perimeter of a ball. You can recover whether the tiling was on a topological cylinder vs. the plane, as you can still define the "end" of a graph, and see that a tiling of the cylinder will have two ways to go off to infinity rather than 1 in the plane.

This is a start of what is knwon as *geometric group theory*. Given a group and some finite set of generators for that group, you can construct a Cayley whose vertices are the elements of that group, whose edges connect an element $g$ with $gg_i$ and $gg_i^{-1}$ for each generator $g_i$. Then you can try to recover information about the group from the geometric properties of the graph.

There is a natural metric $d$ on the Cayley graph, so that each edge has length 1. From one perspective, it's bad that we are getting different graphs from different sets of generators. To identify these as essentially the same, we consider *quasi-isometries,* maps $f$ from one space to another such that there are constants $C_0$ and $C_1$ so that for every $x,y$, $\frac1{C_1} d(x,y) - C_0 \le d(f(x),f(y)) \le C_1 d(x,y) + C_0$. Changing from one set of generators to another is a quasi-isometry, since we can express each generator as a finite word in the other set of generators. Thus, many people study finitely generated groups up to quasi-isometry.

Choices for sets of relations may correspond to tilings. You can attach a 2-cell to the graph along the word of a relation. Topological and geometric properties of this complex have meaning in group theory.

Anyway, back to tilings of the plane. There are more reasons to keep the geometry. This picks out a few graphs among the many which embed in the plane. We also get convenient ways to compare tilings. For example, we can look at the vertices of a second tiling which are near a vertex in the first tiling. We can more easily consider entire families of tilings to try to classify all tilings of a type.

For the Penrose tilings in particular, I would hate to ignore the nice lift from number theory of a tiling to a map from the plane to $\mathbb R^4$. If you consider a tiling by rhombuses so that each edge is $\pm \zeta_5^i$, where $\zeta_5$ is a 5th root of unity, then you see that you can give each vertex a 5-dimensional set of coordinates as an integer linear combination of the 5th roots of unity. Of course, since the sum of the 5th roots of unity and 1 is 0, you can drop the dimension to 4 by considering sums of fifths of integers which add up to 0. There are nice ways to generate Penrose tilings by 2-dimensional planes in that 4-dimensional space. I don't know the classification of Penrose tilings, but I bet it has something to do with that lift, which is not obvious from the graph.