We tend to call all cleansers "soap" so it has become a catch all term. The fact is, though, that soap (originally lye based, made from pot-ash and lard) is very drying to skin and leaves a residue, plus is much more of an irritant to sensitive skin. Most "anitbacterials" have some kind of a soap base. In short, if it's made for hands, don't use it anywhere else! And, face and body skin are different too. So you'll see body wash and facial cleanser. Same for lotion/moisturizer. There really is a difference, it's not just a gimmic to get you to buy more "stuff".
If you have sensitive skin, you want to find high-quality cleaners. I would avoid the grocery store/drug store products (I'm not just talking store brand, but anything you buy at the grocery store, or drug store). They cost less because they have fillers to bulk them up: water, alcohol, oil, frangrance, and other stuff that cost less, take up space, and aren't great for your skin. Generally if you buy it in the mall (cosmetic counters, etc) it's going to be a higher quality, and usually gentler on sensitive skin.
Fragrance can be a big irritant to sensitive skin, too. Especially for your face - pretty much anything you put on your face should be unscented/fragrance free. And watch out for heavy perfumes in body lotion or wash, too.
One great product (with extensive testing for being safe and effective too) that doesn't cost a ton is Mary Kay. Go to www.marykay.com to find a consultant (just enter your zip code) who can help you pick what is right for your skin.