I shop at both and have noted them...
Painfully Full
...and...
Comfortably Empty
...respectively. When I walk out of Asda they don't know who I am or anything about me. When I walk out of Tesco's they know who I am, where I live, what I buy, whether I'm loyal. I might (they may conclude) be an Asda convert who needs tempting back :)
4 weeks of "Spend £20 and get £4 off" has proved irresistable to me. The biggest difference between the two firms is Tesco's loyalty card, and I wonder if it is finally proving decisive?