It seems unlikely this hasn't been asked here before, but I couldn't find a good duplicate.
Password managers, when combined with long, randomly-generated passwords, operate on an approach I like to call "put all your eggs in one basket, and guard that basket really really well". The length and randomness of the passwords protect against brute force and dictionary attacks, and the randomness also prevents password-sharing attacks where one site stores your password insecurely. The primary attack vector, of course, is an attacker gaining access to your password manager database, at which point they gain access to every account of yours.
If you can remember all of these long, randomly-generated passwords, then there is no database to breach, and you're better off. However, most people can't do this.
As an alternative, you can use an algorithm to reproducibly generate a password on-demand for a site (a number of options are described in What is your way to create good passwords that can actually be remembered?What is your way to create good passwords that can actually be remembered?). These retain the benefit of not having a breachable database. The question, then, is to whether or not the benefit of avoiding the database of passwords outweighs the downsides of moving from randomly generated passwords to a reproducible scheme.
If the scheme is obvious upon looking at a plaintext (e.g. siteName + accountName
), then one breach at an insecure site opens up your entire set of accounts to any attacker who happens to look at your password. This is obviously pretty awful from a security perspective. Most schemes aren't that obvious, but you're betting on the attacker to not be able to reverse-engineer it - whether that's a good bet or not is up for debate, and depends on your specific scheme as well. It's worth noting that the chances of successful reverse engineering go drastically up as an attacker gains access to plaintext passwords from multiple sites.
Another issue with reproducible password schemes is that they don't provide an easy way to change your password for a site - they are reproducible, and to generate a new password, you would need to integrate an additional piece of information (some sort of "round"), which you then need to either store or memorize.
Both password manager use and reproducible password schemes are security downgrades from the ideal password memorization option. They are also both much better than what most people do, which is to use the same insecure password across many accounts. Which one is "better" depends a lot on the specifics of your password manager and the scheme you're comparing it against, and which attacks you're worried about.