What is the significance of the graph isomorphism problem?

You are mistaken that there is no known algorithm for graph isomorphism. There is an obvious algorithm: given graphs $G$ and $H$ with the same number of vertices, enumerate all possible bijective mappings from the vertices of $G$ to the vertices of $H$, and then check (in $O(v^2)$ time) to see if each mapping is an isomorphism. This algorithm is guaranteed to terminate; it has running time $O(v!v^2)$.

As I understand it, though, the theoretical significance of this problem is that it is suspected to be NP-intermediate (NPI). That is:

1. It is clearly in NP. (The NTM guesses a possible isomorphism and then checks it in polynomial time)
2. It is suspected not to be NP-complete
3. It is suspected not to be in P.

The existence of an NPI problem is equivalent to P≠NP, so NPI problems are interesting for at least this reason.

Garey and Johnson give the following reasons for suspecting that graph isomorphism might be NPI:

Researchers who have attempted to prove that GRAPH ISOMORPHISM is NP-complete have noted that its nature is much more constrained than that of a typical NP-complete problem, such as SUBGRAPH ISOMORPHISM. NP-completeness proofs seem to require a bit of leeway; if the desired structure $X$ (subset, permutation, schedule, etc.) exists, it should still exist even if certain aspects of the instance are locally altered. Forexample, a function $f$ will be an isomorphism between a graph $H$ and a subgraph of a graph $G$ even if we add edges to $G$ or delete edges not in the image of $f$. However, if $f$ is an isomorphism between $H$ and $G$ itself, then any change in $G$ must be reflected by a corresponding change in $H$, or else $f$ will no longer be an isomorphism. In other words, proofs of NP-completeness seem to require a certain amount of redundancy in the target problem, a redundancy that GRAPH ISOMORPHISM lacks. Unfortunately, this lack of redundancy does not seem to be much of a help in designing a polynomial time algorithm for GRAPH ISOMORPHISM either, so perhaps it belongs to NPI.

(Computers and Intractibility, pages 155–156)

In contrast, the subgraph isomorphism problem is known to be NP-complete. This is the problem of deciding, given graphs $G$ and $H$, whether $G$ is isomorphic to some subgraph of $H$. An efficient solution to this problem obviously solves Clique, Maximal Independent Set, Hamiltonian Cycle, and other similar problems.