Neither is better than the other. They aren't competing, really, but simply different approaches for a problem. Let's use an analogy.
Feet vs Car
Manual Testing (Feet)
Think of manual testing like walking or running. You need to walk places. I mean, even if just to get from your bed to your bathroom in the morning, you need to walk. With walking, it's easy to change your direction or destination. You can do a closer examination of things as you go by, including stopping to smell the roses. And all it takes to do it is your time.
But it's not the most efficient way to travel. It takes a long time to walk or run any significant distance. And doing so also takes a lot of energy. And with it, you're more prone to get lost along the way, especially on large trips. Additionally, if you walk the same route every day, you may not actually take the time to stop and smell the roses and you may miss when the wilt.
Automated Testing (Car)
On the other hand, automation is like a car. It's fast (vrooooom). You still have to drive it, but once you've got it down, it's great for getting you from point A to point B. And it's easy to follow the same route all the time. Plus, it has all these indicators so you can see exactly what might be going wrong.
But you can't stop and smell the roses. And you're stuck on a road, you can just drive around arbitrarily. Also, it's expensive. I mean, have you seen the price of a car? Plus the cost of gasoline. It can get spendy. There are ways to help mitigate that cost (ride-sharing or taxis, aka QA services), but it's still more expensive the walking. And it can only get you so far. You can't exactly drive up and down the aisles of your grocery store, you gotta walk for that.
Why not both?
This either/or approach to manual and automated is very limiting.
Manual testing is fantastic, but you can't rely solely on it for a large project. It's too much work and too much time, and no matter how fast you run, you'll never cover that distance. You need to augment it with automated testing (your car) so you can get where you need to go.
Automated testing is very powerful, but also fairly rigid and expensive. There are some things you can't test with it, and it is only setup to check for exactly what it is looking for, nothing else. You really need to do more than just automated testing and dig into things manually sometimes.
And all of this depends a lot on your project too. If you're going to your mailbox at the end of your driveway, you probably don't need to drive. Similarly, for small projects you may not need much, or any automation. But if you're trying to go across town or across the country, you definitely need automation to help you. Otherwise you're gonna be left in the dust.