A cobbler shaker includes a cap and built-in strainer, which makes it friendly at first. The tradeoff is that it can stick, leak, and slow you down when making more than one drink. A Boston shaker asks you to learn a firm seal and use a separate strainer, but it cleans quickly and works better once you practice. For a home bar you want to keep using, the Boston set is usually the stronger buy.
The Kit / Tools
Boston Shaker vs Cobbler Shaker: A Beginner's Guide
A cobbler shaker is tidy, but a Boston shaker is faster, easier to clean, and better for repeated use.