Pail and bucket can mean the same thing. However, if we want to talk about something of great size, we would tend to use 'bucket'. For example, the indivual seats in the front of a car are called 'bucket seats' rather than pail seats. Also, the huge shovels used in mining are called buckets.

People who don't know any better will refer to a bucket and a pail interchangeably. However, there is a difference! A bucket is normally plastic, although metal ones are available as well. A pail is ALWAYS metal, never plastic. A bucket is used for everyday life, in homes, churches, at a car wash. BUT, a pail is only used on a farm, for milking cows. If you were to say "pail" to a dairy farmer, he would think you meant a container in which you catch milk. Although the two words can be interchanged in regular everyday life, it is important to understand and know the difference. :)

If I ask someone for a bucket, I will get something back in the shape of a bucket or a pail. But I only use the word "pail," for a bucket that is made of metal, such as a sand pail or milk pail. "Buckets" to me are usually made of plastic or wood. Because the terms are often used interchangeably, you might want to add an adjective if you want something specifically, such as "I want a bucket, a METAL bucket if you have one."

Pail is American English. Bucket is more widely used.