A strong approach would be, rather than picking one over the other, to increase the amount of exposure your inventory gets by approaching the scenario from a "why not both?" perspective.
Your own webstore, of course, gives you a lot of control over branding, and how you run your business. But this means, as Andy pointed out, that you're solely responsible for marketing yourself and building your customer base. This can take a long time and be super involved.
A marketplace like eBay or Amazon, however, has MILLIONS of built-in customers, already looking for whatever it is you're selling. This is why they charge fees the way that they do.
When you have both, however, you're able to give your inventory that much more exposure. Software like Solid Commerce can help make it easy to manage things like inventory and listings when you're selling on more than one channel.
Solutions like SC make it possible for sellers to avoid feeling trapped by one option or the other, and instead spread their inventory around the web as much as possible, for maximum exposure.