If you look in the Corpus of Contemporary American English (COCA), you will find that different than, despite objections to its being “illogical” and “incorrect”, is in fact a very common and therefore standard usage in American English.
|DIFFERENT FROM raw count
|DIFFERENT FROM per million
|DIFFERENT THAN raw count
|DIFFERENT THAN per million
Raw results page
From this we can see that, indeed, different from is more common than different than, but by a ratio of less than 4 to 1, meaning that different than enjoys substantial minority usage. Further dividing up the usage, we see that different than is almost as common as different from in spoken English (22.61 incidences per million words for different from versus 19.81 for different than), but much less common in written forms. When a usage is more common in spoken English, that is usually a sign that it is less formal.
For another perspective, let’s look at the historical development of different than using data from the Corpus of Historical American English:
COHA results page
Here we see that different than is a relatively new development in American English, only coming into any significant usage starting in the 1960s. It is probably this relative newness that makes usage commenters object to different than. But the rise of different than is probably inexorable, and the COCA data, which divides up incidences over the last 4 half-decades, shows that the ratio in favor of different from was 4.4 to 1 for 1990–1994, but had dropped to 2.9 to 1 by 2005–2010.
So, in conclusion, yes different from is more common than different than, and different than is less formal than different from, probably because it is a relatively recent development. However, different than occurs with significant frequency even in formal academic writing, so to write it off as simply “incorrect” is to ignore the facts. If current trends continue, different than and different from will be equally common within a few decades.