The importance of the melting glaciers is clear for countries like Switzerland, which are landlocked and use mountain runoff for freshwater. The melting glaciers provide a lot of fresh water to streams and rivers, that is then used for the human population of those countries. In a glacier-free Europe, that source of water is gone, and countries will have to rely solely on seasonal snowpack melting. This isn't a great strategy, though, since the melt season is getting longer and longer, and seasonal snowpack is quite variable. Essentially the absence of the glaciers will add a level of instability to the water resources of interior Europe.
It is also worth noting that this story comes on the heels of a scientific paper about changing streamflow made waves two weeks ago. The paper is by Dai et al., and is in the upcoming issue of Journal of Climate. Some of the news coverage of the paper was a bit out of control. The paper is a compilation of data sets from around the world, measuring streamflow in large rivers. The main point of the paper is to show the basic results of the combined data set, and to provide a climatology to other researchers who want to know how much freshwater is going from the land into the oceans. You wouldn't have gotten that from the media coverage, much of which seemed to focus on the Colorado river, which isn't even mentioned in the paper by name. The headline grabbing aspect of the paper is a cursory examination of the trends in the data. Freshwater input into all the ocean basins except the Arctic show a decrease in overall streamflow, while the Arctic has an increase. It is not clear from my reading of the paper that these trends are (1) statistically significant or (2) attributable to climate change. The first point will be cleared up over the next couple of years as more people look at the statistics of this and other data sets. The second point will hopefully be better accounted for in future studies by trying to explicitly incorporate human-made changes in streamflow (for agriculture and including building dams and resevoirs). In the meantime, it is an interesting data set, and could have important implications for budgets of freshwater flux to the oceans. Whether the rivers are drying up remains an open question, and one that a lot of us are very interested in.