If you are knitting in the round, you are knitting with your stitches joined and producing a tube. If you are using the circular needles as straight needles and knitting flat, there should be no difference from when you use straight needles.
I think that this pattern will produce something like a seed stitch. As long as you are following the pattern as written it should work out. Different yarns will produce different stitch definitions and if you are using a different yarn, that could be part of why it looks different. Could you possibly attach a photo of your knitting? That could help figure out what, if actually anything, is wrong.
Happiness is catching that dropped stitch while it's still fixable..