Yes, sir. You'd be looking at a bit less if you gave the -3 vs. if you took ATL and got the +3. That is of course, under the assumption that there are other sports books who don't have the juice as -110 apiece.
On my book (largest online book), you can give the -3 and take the Pats, but it will cost you somewhere around -120 or -125 (has been varying a bit today).
In comparison, taking ATL+3 still allows you to bet with the usual juice, so technically you'd make a few more dollars than if you bet NE.