Depends on a lot of things. Bar feeders and parts catchers make for great automation but they have limitations. For larger parts, the parts catcher might not have enough outfeed capacity to be very useful for unattended production. Lots of different types of parts catchers out there. Bar feeders are generally best for 2.5" diameters and less. "Part integration" time for a bar feeder (time it takes to get it working the first time) is generally fast for bar feeders. "Part changeover" time is generally under an hour for bar feeders
Robot arms can handle both small diameters and much larger diameters but are limited by the payload capacity of the robot. A dual-headed gripper (unload finished part, load next raw material without going back to infeed) will increase efficiency a lot. Lots of choices in infeed/outfeed with robots. Consider what you plan to run over the life of the system. If you only plan to run short parts in a "puck" configuration, a drawer or stacker infeed can be a good choice. If you need to run "shafts" or taller parts, a table infeed provides a lot more flexibility and can often achieve infeed densities equal to or greater than drawers or stackers by stacking parts in the infeed.
There are big differences between robot systems in terms of "part integration" time and "part changeover" time. On the mechanical side, what needs to be engineered to add a new part? Gripper fingers? Part locators? Those can add cost and time to part integration. Some systems require a lot of engineering for part integration, others require none.
Some robot systems are pre-programmed. Just fill out a form, save the configuration, validate the configuration, and you're ready to call up that job anytime. Typically less than an hour for part integration. In other systems, you are on your own and every part integrated is a new robot program. If you're a solid robot programmer starting from scratch, first part integration might take 1 to 5 days, subsequent parts less. If you're not a competent robot programmer... I come across many robots sitting in the corners of shops collecting dust.
For part changeovers, look at how much needs to be changed during a part changeover. Some systems only take a few minutes, and others can take a few hours up to a day or more.
In general, lots of small parts, with small diameters, bar feeders work well. With more variety in part sizes and shapes, a robot system will typically work better. Just make sure you know what you're getting into with the robot system and if it is really designed for the type of production you're doing.