It could be due to difference in the speed ratings, tread pattern, tire compound, and/or traction rating. I think the issue is the difference in speed rating - Dunlop SP Winter Sport M3 tires are H or V rated "Performance Wintern/Snow"; Blizzaks "Studless Ice & Snow".
Bridgestone:
Dunlop
I doubt it's an alignment issues unless you were having issues "drifting" before, or recently hit something (curb, rock, horse, etc...)
Edit: Note that you're not going to get the same "driving on rails" traction as a set of dedicated performance tires from a set of all-season tires, and especially not from a set of snow tires. It's generally a trade-off.
2nd Edit: If you were advised to buy these tires from a particular shop/dealer, you should return and demand another set. It is not advisable to mix/match tires of different speed ratings, much less different categories altogether. It's not safe and from my own experience, I would generally not sell a customer tires if that was the only option we could provide - I'd sooner refer them to another shop that had the correct tires, or advise he/she change all 4.