A "spelling bee" is an expression usually used to describe the spelling contest, but not necessarily. With that in mind, is it be considered redundant to call it a "spelling bee contest" or a "spelling bee competition"?
In the US, the term spelling bee is very common, but it is rarely seen or heard immediately followed by the terms contest or competition. This ngram shows that the phrase standing alone is nearly a hundred times more prevalent.
Most US speakers would consider the terms contest or competition to be redundant.
There is an entry in the OED which explains the term originated in the US farming community, where groups of neighbours would get together to help one another. They would, for example, hold 'an apple-bee' to gather in the apples; a harvest-bee, a quilting-bee etc. The term was an illusion to the social character of the insect. Spelling-bee derived from similar groups getting together to hold a spelling competition.
I have no idea if they still have them in the USA, but it is certainly a dated concept in Britain. Though I guess what we are doing here on this site might be termed a 'literacy-bee'.