Any packed paper is difficult to separate because air has been compressed out of the stack and a static charge holds the sheets together. Shuffle a new deck of cards and it will sit higher than a new unshuffled deck because there is more air between the cards and they are easier to deal because the static charge has been reduced.
The trick is force more air between the bills. Something as simple as folding over a corner of your stacked bills will introduce enough air at the fold to allow them to be more easily separated. Folding also push the bills off-center of the stack, leaving a slanted edge which is easier to manipulate. Crumpling new bills works not because the bills are crumpled but because the crumpling allows more air to remain between the bills.
Tackiness works well with some types of paper, but currency is made to resist tackiness. Air is your best weapon against “sticky” bills.
People seem to intuitively know this because you’ll often see people do strange things with stacked paper. Have you ever seen someone riffle a ream of paper before putting it in the printer or copier? Introducing air helps release static bonds which make it less likely that the machine will pull through more than one sheet at a time.
People who work with new currency a lot may blow along the long edge of the bills or fold a stack of new bills lengthwise. They may not be able to tell you excactly why they do these things but there is only one reason to do it - to introduce air.