They don't spend much time in developing traditional algorithms anymore (e.g. search engine). They don't reinvent the wheel, they re-use existing algorithms, even though they could do a better job by re-inventing these old algorithms. But they think at significantly improving existing algorithms (e.g. search engine), both from a business and technical point of view. Sometimes they even do a bit of hacking and reverse engineering.
They spend a lot of time identifying useful data sources (most of the time, external sources), and checking how they could be exploited and leveraged.
They develop meta algorithms: that is, algorithms that rely on lower level techniques such as taxonomy creation, spell check, keyword associations, news feed aggregation etc. They blend these low level algorithms to produce automated high end products.
Modern data mining is to old data mining what coding in Python is to coding in assember, Perl, or C.
These data miners don't call themselves data miners anymore: they are statisticians, scientists and data miner all at the same time, combining all these roles in one person. Indeed, they are even business people and sales guys (selling internally - to their boss and employees reporting to them, and externally - being invited at important sales calls by their company).
Many times, they succeed not by analyzing data, but by applying analytic thinking to business problems. Example: we've increased our signups by a factor 3 on our network thanks to analytic thinking - however no data was leveraged to get to the action resulting in signups improvement.
They sometimes develop great products without having to produce one line of code: our news feed optimizer is a good example of a technology that was developed without writing any code, but instead relies on widgets to distribute the news, manual selection of good feeds, and intelligent use of feed aggregators such as twitterfeed or feedburner.
They find the information they are looking for on the web (and more and more from social networks as opposed to Google), but not from books nor University training.
They are working on algorithm speed optimization and scalability, even (and especially!) when developing prototypes.
They are thinking so much out-of-the-box, that they are not generally a good fit for the corporate world. Instead, they become successful entrepreneurs.
They are domain experts more so than coding experts. But they are also generalists across many fields and many tech skills (e.g. all of them are very familiar with SQL and R). At the same time they have deep expertise in a few (one or two) specialized domains and programming languages.