The problem with using Erdos-Renyi random graphs ($G(n,p)$ or $G(n,m)$) is that they follow a Poisson degree distribution, which gives them finite second moment. Many real world graphs, including the "Web graph" or the "Internet Graph", tend to not follow this degree distribution in favor of a degree distribution that has much more variability in it's second moment. In my opinion, one of the biggest differences is the power law degree distribution that many of them have. See Emergence of Scaling in Random Networks for example.
As you probably know, there appears to be a difference between the connectivity graph for the World Wide Web and opposed the connectivity graph for the Internet infrastructure. I certainly don't claim to be an expert, but I've seen Li, Alderson, Tanaka, Doyle and Willinger's paper "Towards a Theory of Scale-Free Graphs: Definition, Properties, and Implications" who introduce an 's-metric' to measure the 'scale-freeness' of a graph (with the definition of scale-free graphs still under debate as far as I know) that claim to have a graph model that creates graphs that are similar to the internet connectivity at a router level.
Here are a few more generative models that might be of interest:
Berger, Borgs, Chayes, D'Souza and Kleinberg's paper "Competition-Induced Preferential Attachement"
Carlson and Doyle's Highly Optimized Tolerance: A Mechanism for Power Laws in Designed Systems
Molloy and Reed's A Critical Point for Random Graphs with a Given Degree Sequence which introduces the "Erased Configuration Model"
Newman's Clustering and preferential attachment in growing networks (which has been mentioned already)
One could also explicitly generate a degree distribution and create a graph this way, but it's unclear to me how close this models the internet graph at a router level.
There is, of course, much more literature on the subject and I've only given a few of (what I consider to be) the highlights.
As far as I understand, many results that worked for the Erdos-Renyi models of random graphs ($G(n,p)$ or $G(n,m)$) do not work precisely because the scale-free or power law degree distributed random graphs diverging second moment in the degree distribution. I don't claim to know enough about to the subject to categorically make claims about "most" proofs, but from what I've seen, one of the first few lines of proofs for properties on Erdos-Renyi random graphs explicitly assumes a finite second moment in the degree distribution. From my point of view, this makes sense as a finite second moment makes Erdos-Renyi graphs much more locally tree-like (see Mertens and Montanari's Information, physics, and computation) which effectively gives properties/paths/structures independence. Since power-law degree distributed random graphs have a diverging second moment, this local tree-like structure is destroyed (and thus requires different proof techniques?). I would be happy to have this intuition invalidated if someone with more knowledge or insight were to show why this is not so.
Hope that helps.