I think the difference is in the aims. With traditional statistics, you're trying to estimate some quantity of interest in the population, while with "big data", you're typically trying to make predictions for individual users. While this can be done with traditional statistics (in fact, the predict method in R does exactly that) it becomes easier to match participants on what books they might like if you have data for what books everyone in your population likes rather than just a sample.

Now, whether or not the inferential premises of statistics hold up on website data (and population data) that typically is neither random nor representative, that's another story.