I would say 'pail' is a bit old fashioned and 'bucket' is definitely used more often. For example " Jack and Jill went up the hill to fetch a pail of water..." this is a very old nursery rhyme.

In the US, in most regions, we say bucket. There is no real difference between pail and bucket, but sometimes buckets smaller than 5 gallon/20L may more likely be called a pail. Most buckets/pails you come across in the US are the 5 gallon variety. Bonus: in the US, we often refer to shipping 'pallets' as 'skids.'

A bucket and a pail are both containers typically used for carrying and holding liquids, sand, or other materials, and they are quite similar in function. However, there are some regional and historical differences in the usage of these terms. Bucket: The term "bucket" is more widely used and recognized, and it is used in both American and British English. It's a generic term for a cylindrical or semi-cylindrical container with a handle that is used for various purposes, such as carrying water, cleaning, gardening, and construction. Pail: "Pail" is a less commonly used word in modern English, and its usage is often associated with older or more traditional language. While it's still understood by most English speakers, it may sound a bit more old-fashioned or formal compared to "bucket." "Pail" typically refers to a bucket with a flat bottom and straight sides, often made of metal or plastic, but the term can be used interchangeably with "bucket" in many cases. The preference for one term over the other can be influenced by regional variations in language and personal preference. In some regions, "pail" may still be commonly used, while in others, "bucket" is the more prevalent term. It's also worth noting that the choice between these words can depend on the context and the specific item being referred to. In everyday casual conversation, "bucket" is more commonly used and understood by a wider audience.