There are no silly questions...
When using a cable STB, there is no "scanning" needed by the TV set; all the tuning functions are done in the STB, and the signal is output as an ordinary analog NTSC (480i) signal on channel 3 regardless of what the nature of the selected channel is. If the TV is connected by a single coax to the STB ("cable out") using screw-on (type F) connectors, standard definition analog NTSC on ch 3 is all you will ever see.
If the box is not really ancient, it will ALSO have as suggested above a composite analog video output (a yellow RCA jack) along with stereo audio (on a red/white pair of RCA jacks). To use this set of outputs (usually just called "composite") to connect to an A/V input of the new TV set would be a small step up in quality but still give you only analog NTSC video and stereo audio.
An entry-level digital STB (no PVR function) will ALSO have a component video outpt (3 RCA jacks, red, blue, green), digital audio, on either an orange RCA jack or an optical jack (square), AND probably an HDMI connector (rectangular multipin). If you could connect your new TV to a digital STB using either component or HDMI cables, you would be able to take advantage of the much better quality standard definition digtal signals, as well as some high definition channels, depending on how your provider tiers and prices "digital" services.
You didn't say what the model of the "old" cable set top box is, and that's important. If it's an older analog-only unit, you can't get any of the benefits of your new TV. Most CATV cos charge the same fee for a non-PVR settop box itself of any flavor, so you might consider swapping the old one out for the HD digital version. Unless you add new services, your bill shouldn't increase, however many CATV providers may "require" you to add a digital tier before they will lease you a digital STB. Ask politely in person at your provider's local business office to find out what their local policy is.
Ultimately, if they won't make the STB trade at no price increase, I believe you will probably find that the lowest digital tier would be well worth the additional cost per month and make your new Sharp TV really come alive.
Hope that helps...