Did Target send you a letter saying Equifax FICO score?
The scores on Equifax website are FAKO
OP, if you mean your score from myfico, it's a version called Beacon 5.0, that's commonly used for mortgage apps. The score commonly sold directly from the equifax site is indeed a fako.
I'm not familiar with the "redcard," so I don't know the fine points....(others that have it will jump in eventually).
It's possible that Target gave you either a card enhanced score, or an internal score, which means any past bad cc habits will affect it accordingly.
+1, the EQ fico provided here will match what a mortgage broker would pull if both are pulled at the same time. When I applied for a mortgage in January, the EQ the lender pulled matched exactly with the EQ score provided here. Credit card companies often use a different version or they have some internal scoring algorithm that they use.
Cards: Orchard Bank ($1100) | Cap1 Cash Rewards ($2500) | Chase Freedom ($1000) | Best Buy ($2500) | Discover It ($1000) | Barclay Rewards ($2500) | Current scores: EX FAKO: 684, CK TU: 649, FICO EQ: 680, FICO TU: 698, FICO EX: 658 Happy Homeowner Since 2/6/13! 
Last App: 4/5/13 Gardening until July 2014
What's your AAOA, and utilization?