The "pocketed" 2x2 (and other sizes, as Sap aptly points out) are generally used as a temporary storage medium.
The EAC (Early American Copper) recommends 2x2 paper envelopes with a soft cotton liner. Much more difficult to use as the coin must be removed from the envelope then the cotton liner bag, before it can be studied.
The plus side to the pocketed 2x2's is the amount of information one can write on them, same goes for the envelope. As for Ancient collectors, I am more accustomed to seeing these coins in the paper envelopes with the pertinent information written on the outside. Can't say I've seen many ancient dealers or collectors using the cotton liner bags though.
Sap also made a very valid and important point that the softer flips probably contain PVC plasticizers in them, which can cause damage to coins if stored in them for extended periods, they tend to outgas and release a hydrochloride gas that can turn into acid which will eventually etch into the surface of the coin stored in them. The PVC flips are used by dealers as they are soft, easy to put coins into and remove, and don't crack or scratch the coins high points with hairlines. The major grading services like PCGS, ANCAS and NGC all request that they receive coins in the large (2.5" x 2.5") soft PVC fips as they are safe for short term storage and are easy to get the coin out of when time for grading.
That said NEVER, NEVER, NEVER store your coins in the soft flips for any extended time (more than a month or so) especially in a humid and warm environment. This will accelerate any damage that could occur.
The harder flips are usually safe and will say so by each manufacturer, look for Saf-T-Flip, or archival quality labels. The big downside to these types of holders is they are tough to get a coin into and out of (I often ruin my cuticles using them, blood on coins is probably not a good thing), they can cause scratching/hairlines on uncirculated coins, the flips are brittle and can crack, they scuff easily over time and make for an ugly holder.
I personally use the cardboard with mylar flips stapled (many collectors hate these for that reason) staples can and do scratch coins, I use the flat cinch stapler now, but for years I used a pair of needle nose pliers and flattened each staple carefully by hand. They adhesive type I've never liked as I don't want adhesive near my coins, and I also worry that over time the adhesive material will outgas and could potentially damage my coins. When I remove coins from the cardboard flips, I use a razor blade to cut the around the mylar circle to release the coin over a padded jeweler's tray. Most staple scratches are caused by novices pulling the cardboard flips apart and sliding the coin out.
There are many other types of coin holders made also, Capital Plastics makes plexiglass holders where the coin sits in a hole and is sandwiched by clear sheets of plexiglass held together with plastic screws, There are the whitman hard plastic snap-lock holders (I find coins can rattle around in them, and they scuff up too easily on the outside),
Coin World offers a small slab type holder that snaps together. Intercept holders another plastic snap holder in 2x2 sized square with an inert liner cut to fit various sized coins. Lighthouse (usually found in Europe more often than the USA) makes various lines of similar high quality holders, many of which are designed to snap into pages or albums they also make (expensive). One can also purchase small capsules (air-tites) that cover a coin in a round shape, some fit the coins directly some have sized inert washers to hold the coin.
One last trick I've used with the 2x2 flips successfully (as do many dealers and
TPG's), is to get small poly bags (usually 1.75" wide by 3" long, and drop the coin into them, before putting them into the soft PVC flips, this offers protection against outgassing and PVC damage over time.

We haven't even touched on various albums yet.

"Buy the Book Before You Buy the Coin" - Aaron R. Feldman - "And read it" - Me 2013!
ANA Life Member #3288 in good standing since 1981, ANS, Early American Coppers Member (EAC), Colonial Coin Collectors Club member (C4), Conder Token Collector Club member (CTCC), Civil War Token Society (CWTS) member, Liberty Seated Collectors Club (LSCC) & Numismatic Bibliomania Society member (NBS), USMex, Member in good standing, 2¢ variety collector.
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