There is some confusion here about the two alloys. Hopefully this will benefit others as well.
"6063" and "6061" are not sufficient descriptions of the materials in question. To know the specific differences, you need to know the specific heat treatment/tempering process of the alloy.
If you compare 6063-T6 and 6061-T6 you will find that their properties are similar with 6063 having a slightly lesser Ultimate Tensile and Yield Strengths. However, if you look at mcmaster, I can only assume they are selling a 6063-T4 and 6061-T6. If you compare them, then yes, 6063 has a considerably lesser ultimate tensile and yield strength. However, for 6063 the % elongation almost doubles, from 12% to 22%.
The machnining differences have everything to do with the heat treatment as well. The T6 alloy will probably machine better than the softer T4.
Edit: Seems that kevin posted part of what I was getting at before I was finished writing.
I will leave it as is with a little extra explanation.