I've tried both methods, trying to find the least fatiguing way to hand quilt. For both methods, it's mostly about what the hand that is not holding the needle is doing. With the Thimble Lady's method, you are folding, or pleating the fabric onto the needle. With Aunt Becky, your needle strikes the metal, you know it has made it through the backing and can then deflect it back up to the top. With both methods, you control the needle with the dimple of your thimble.
I used the Thimble Lady's method for a baby quilt and was able to get smaller stitches than regular quilting, but it took really close basting for me to keep the layers smooth. A shorter quilting needle was easier for me to control than one of her longer needles.
Aunt Becky seems to keep your hands in an even better position for long quilting sessions, but I'm still trying to get my stitches smaller. I haven't found the rhythm of moving Aunt Becky at the exact moment to get a small stitch. So far, I can get even, but not small-- thinking it's too thick for me. I'm going to try the same method, but use a thinner device, so I'm going to buy a banjo pick and plan on trying to flatten the top of an old thimble that isn't as deeply dimpled as new one. I also tried an old silver demitasse spoon, but it wasn't angled right. If I can't get any of these to work for me, I'll go back to Thimble Lady.