Learn English – the difference between “park” and “parc”

I feel like the word pail almost always describes a metallic object, shaped in a near-cylindrical fashion. Sometimes a pail can be wooden, but rarely. Buckets can be made of any old material, especially plastic, and can be shaped more strangely than pails.

In addition, bucket has some interesting and amusing uses in slang:

1. In its plural form, it can be an expression of unalloyed happiness. It comes from the slang term from having just scored a field goal in basketball. For example, if you had just won something unexpected in the mail, you might say "Buckets!" to celebrate it, just as you might having scored playing basketball.
   

2. It can describe a particularly decrepit vehicle, a hoop-ti; most often applied to vans.

3. It's an urban slang term for urban-style hats, typically wide-brim and loose fitting.

4. It's an urban slang term for expensive rims on a car. As so memorably used by the rapper Yung Joc, "...ride around slow so you can see the buckets on my feet [tires]..."
   

Pail, sad to say, is utterly lacking in this regard.

EDIT: Taking a look through Google's N-Gram viewer, it's not hard to see why:

This comparison of bucket and pail from 1800 till today shows the latter's usage diverging noticeably from the former's around the era of 1940–1960, to becoming a much less popular a synonym for the former nowadays. The chart makes a lot of sense to me, at least superficially; the 1950s–1960s was an era where college attendance and job mobility were first greatly expanded and democratized, and where a lot of young adults who might have grown up to work on the family farm in older times instead found white-collar, professional work. As pail in literature is strongly associated in my mind with farming contexts, it makes sense to me that authors would have limited their use of pail in that era given its more limited relevance towards their target audience. If a word doesn't quite have a "regular" currency, obviously there will be fewer opportunities for it to make its way into slang usage.

There are many cases where either word could be used. However uncover carries the implication that something was covered or hidden, often on purpose or perhaps with the passing of time. Something that was previously known is being re-discovered.