A. Buckets are cheaper. A super nice bucket with handle and a lid may run you about $10, if you're getting the short end of the stick. A briefcase on the other hand will run you $30 for one that will last about 2 uses, and up to I would guesstimate about $100k for an Italian Leather one, made out of real Italians.
B. Obviously buckets hold solids, but they can also known to few, carry liquids. Now these liquids can be in container themselves, (when was the last time you asked for a Briefcase of Beers?) or just straight take the form of its container. I tested my hypothesis about a briefcase holding liquid as well... and while it can hold liquids, it is tough to keep your job when you offer your colleagues a briefcase of coffee.
C. Buckets have larger volumes per unit surface area. This is a win win to the consumer and producer, the producer has less materials cost, and the consumer gets more utility. While you may be able to produce a flat surface for cheaper than you can a curved one, you can always design a better machine. Try to engineer a better square. Well actually someone did this one time, and it was called a circle.
D. Finally you turn a briefcase upside down and you know what you have? An upside-down briefcase. You turn a bucket upside down and know what you get? A chair? A drum? A cover? A pet bucket?
So my question is, at what point in time did people start carrying briefcases, and why?