When Mikael Jansson defected from the Center Party and took over Sweden Democrats in 1995 he began a purge of the openly extreme-right elements of the party in order to turn it into a respectable anti-immigration movement, but it didn't really solidify until the past few years. Even in recent years they have had a struggle to keep those types out of the party, for example their youth leader from 2007-2010 (Erik Almqvist) was embarrassingly videotaped singing white-power music at a retreat, and was finally forced out of the party earlier this year (weirdly, for someone who opposes immigration, he reportedly emigrated to Hungary after resigning).
DF by contrast has different origins, culturally rooted in something more like old-person Danish culture, rather than youth radical-right culture. They have a vaguely mythologized idea of 19th-century or early-20th-century Danish culture, as they imagine one's grandfather in a small town in Jutland might have lived. They generally therefore view the radical right-wing scene as negative, as equally a deviation from traditional quiet Danish values as immigrant crime is. This has let them keep those elements out of the party more credibly, because it's not only for image that they do it, but because their base actually culturally dislikes the far-right (especially anyone associated with the metal or biker scenes, who they see as culturally un-Danish).
Not that I myself would vote for DF (as an immigrant in Denmark they aren't out to make life easier for me), but I somewhat understand their appeal better; they are more of a rural "traditionalist values" party than a radical nationalist party. I find it harder to understand how SD would get similar support, unless people either don't realize their association with the far-right subculture, really believe that their transformation is completed, or are just lodging protest votes.
Because in Denmark, immigration can be part of public debate without being shunned, while in Sweden, it's hard to get any sort of ground in this debate without being labelled a racist.
So now, some of them are trying to be more 'reasonable' about it with the party Sverigedemokraterna. I assume because they haven't made any progress publicly in Sweden by just being seen as extremists. And more and more figures are beginning to tell Swedes that immigration is not an issue you can ignore, thus Sverigedemokraterna's support is rising, because it's the only party that wants to talk about it.
But I am confident, that most of its votes are 'protest votes', to get the other parties to talk about it.
I will say that the Swedish population has become more polarised over this question - both the number of people who claim "immigration is good for Sweden" and "immigration is bad for Sweden" has risen over the past 10 years, shrinking the middle. There is also a strong correlation between education and attitude to immigration (in 2011, 56% of the people who had only 9 years of education agreed that reducing the number of asylum seekers to Sweden is good while only 22% of those with a university-degree agreed with the statement). This might explain some of the frustrations showcased by the people supporting Sverigedemokraterna and other anti-immigration parties - they might simply lack the ability to debate in the way public debates are typically held and coherently express their ideas in a convincing way.
The lack of immigration debate in the 1990s is what caused Dansk Folkeparti to gain enough support, that from 2001 to 2011, they could control the Danish government. So even if Sweden's immigration debate is happening, immigration debates always happen too late.
(But this is not unique to immigration.)
Sverigedemokraterna is big because a large slew of the Swedish population is slanting toward racism and are prone to accept simple "solutions".
There are also other factors like that some people genuinely seem to believe that SD has cleaned up their act.
EDIT: I am personally strongly pro-immigration (much more so than any party in Sweden) and I have to disagree about there being much debate about immigration in traditional media. There is a lot in social media though, so I have no idea where all the racists get their idea that they cannot debate immigration issues anywhere.
DF really do seem to be critical towards "ethnic Danish" subcultures they view as deviant as well, especially in the past year. There's currently a bit of hand-wringing over biker clubhouses (residents don't like having one nearby), and they've been getting out in front of that populist backlash too, which plays well among their base for different but related reasons. It's sort of Public Enemy #2 in things that are ruining Denmark, from their perspective.
Here's a recent story (via Google Translate, note that "rocker castle" means "biker clubhouse"): http://translate.google.com/translate?sl=auto&tl=en&js=n&pre...
