Yes, the two words "sheep" and "cheap" sound different to native English speakers, and I'm going to see if I can prove it. I'm going to turn on the voice dictation feature on my computer. I'm going to speak individual, isolated words so that the program can't get any cues from context.
Here's what I'm going to say. We will see what the program hears.
Sheep. Cheap. Cheap. Sheep. Ship. Ship. Chip. Chip. Witch. Wish. Witch. Wish. Mash. Mash. Match. Match.
OK, here is exactly how my computer transcribed it. My apologies for the rude word.
Sheep cheap cheap sheep ship shit chip chip which wish which wish Mitch mesh Match Match
The program correctly identified the <em>sh</em> or <em>ch</em> sound fifteen times out of sixteen. It only made one mistake, hearing <em>Mitch</em> (with a <em>ch) </em>when I said <em>mash </em>(with an <em>sh)</em>.
Incidentally, it isn't really a mistake that the program transcribed "witch" as "which." The pronunciation of "witch" is identical to the most common pronunciation of "which." Some very careful (or old-fashioned) speakers will aspirate the first "h" in "which"--it actually is more like "hwich" than "which." There are many bad puns playing on the fact that the two words can sound identical.