Both picture and srcset are not 100% compatible with all browsers, but they both degrade gracefully. If a browser doesn't understand the <picture> element, it will gracefully fall back to the <img> element inside of it. If a browser doesn't understand <img srcset...> it will fall back to using the src attribute of the image.
The <picture> element (and <source> sub-elements) are the heavy guns you bring in when you want to do art direction on different sizes / aspect ratios of the image. The img srcset attribute is much more lightweight and is all you need if you want to design for different resolution displays.
Because they both have measures for backwards compatibility, I would not worry too much about which one you use - both will fall back gracefully in older browsers. If you're only designing for pixel density, I would recommend srcset because it's more lightweight.
