HD pictures are sharper. It's not always easy to tell the difference if you don't have SD and HD screens side-by-side, especially with smaller TVs. I have found that it is more noticeable with live events like sports.
One difference is in the number of "lines" the TV draws, counting from top to bottom. SD signals have 480; HD have either 720 or 1080, depending on the signal coming from the station.
If you are using an X1 box, it will normally auto-tune to the HD version of you try to select an SD channel. Look carefully at the banner when you change, as it will say something like "Auto tuning to channel 1477".
And, of course, you must be using an HDMI input, and the cable box is set to output 720p or higher.