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 cobbler shaker wins the first five minutes. It looks complete, the small cap can measure in a pinch, and the built-in strainer makes sense to a beginner. The problem appears after use. Sugar, citrus pulp, and cold metal can lock the lid in place. The built-in holes are often slow. Cleaning three pieces takes longer than rinsing two tins.
A Boston shaker is less intuitive but more durable as a habit. Two tins seal with one firm tap, open with one controlled strike, and leave enough volume for proper shaking. You do need a Hawthorne strainer, and you need to learn not to overfill it. Once that is learned, the Boston set is faster for margaritas, sours, daisies, and any drink with juice, egg white, cream, or muddled fruit.
For a first purchase, choose weighted stainless steel tins that feel balanced, not decorative. Avoid glass-and-tin sets if you are nervous about breaking glass or if you shake near stone counters. Keep the cobbler only if you make one drink occasionally and value compact storage over speed.
Further reading: Serious Eats tested Boston shakers, Decanter on home bar tools, and background on the cocktail shaker.