Is there a legal or practical difference between a union or association? Specifically, is the Major League Baseball Player’s Association considered a union?
Are there bargaining rights that a union has that an association doesn’t have?
Yes, there is a difference. In the US, unions are overseen by the National Labor Relations Board. Most importantly (although an oversimplification), unions have the right of collective bargaining, the right to bargain with an employee on behalf of its members as group. Generally, to have a union you need to get a certain percentage of employees (35% I think) to sign cards saying the want to be in a union and then file a petition with the NLRB who will hold a secret ballot election of the employees to establish the union.
The MLBPA is a union. It was originally organinzed with the help of the United Steel Workers Union. Here is a brief history of the MLBPA.