I have a 2016 Specialized Tarmac Expert with a Fulcrum Racing S-19 light wheel-set and 700x24mm tires. Do I have to replace with 24mm tires? Or could I use 23mm/25mm?

Any help would be appreciated, tires (even their spelling 'tyres??') can be confusing.

You'd be fine with either 23 or 25.

Great! So many more options then, and I appreciate the quick reply.

Sigh...

Specialized and their weird tire sizes! My Roubaix came with 700x26 tires, but I'm now running 700x25s. Honestly, the numbers are misleading since the current tires blow up wider than 25, and the 26s seemed narrow.

I don't think Specialized holds any less weirdness that others. The size of your rims makes the most difference. You didn't put your tires on the same standard that the mfr designs them for. I'm not entirely sure that every mfr bases the width on the most exterior layer. It might be the width of the casing or another layer.

rim width's also play heck with my perception. The wider rims on new road bikes make 25 and 28 mm tires look like 18mm tires that I used to run on my 14 mm rims.

Well, 24mm and 26mm are much rarer than 23mm, 25mm, and 28mm tires. It would be like someone making a 31mm.