Hey! It's only handlebar tape. Give the cotton tape a try and decide for yourself whether or not you like it. The worst thing that can possibly happen is you'll have to put out $10.00 for new cork tape. Think of it as the price of education.
Cotton handlebar tape will last more than a season (unless you "drop" the bike and the bars take part of the impact) ... particularly, if you wear gloves. You could, if you wanted to, leave it on for many seasons ... and, when it fades, you can unwrap it and rewrap it with the "inside" facing out, or let it age gracefully.
Well, I'm not exactly sure what you are asking ...
The cloth tape provides a GOOD gripping surface -- cold bars in the morning & hot bars in the aftenoon will be buffered ... close to zero padding. Cloth tape is much easier to install ... impossible to tear UNLESS you have Super Hero powers. The dye in the cloth tape may fade over time, but it is theoretically reversible (simply unwrap & rewrap).
The "cork" tapes also provide a GOOD gripping surface AND are thicker ... and, can tear during installation. Theoretically NOT reversible due to the chamfered edges (i.e., for cosmetic reasons) and/or because of the adhesive band [some have it, some don't]. Color won't fade. Water resistant.
Whereas old cork generally just looks dingy or neglected, old cloth tape MAY (or, may not!) have character.
You've got the cloth tape ... there is little lost in using it at this point. You MAY like it better!
Cotton tape is often shellacked. The more coats the more grip/padding. Clear shellack maintains the color more or less, amber shellack changes the color. See Rivendell site (previously mentioned) alao Lovely Bicycle! and Velo Orange. If you want retro/old school/traditional, check it out.