Is there a significant performance difference between the Computed Field module and writing code? When I got started using Drupal 7, I added several fields to my site using the Computed Field module (to calculate age, zodiac sign, and so on).

Now that I have been using Drupal 7 for a few years, I realize that it is generally not a great practice to write PHP code in the UI, as the Computed Field module requires.  But, the code I am using has worked for the past few years, so that is no longer an issue.

However, I am wondering if there is a significant performance difference between using the Computed Field module (saving the result in the DB) vs. running the PHP in hook_node_presave() or in a similar fashion somewhere in a custom module.

Would removing my computed fields (and uninstalling the Computed Field module) and replacing them with fields in a custom module significantly increase performance?
"Significant" really depends on too many things - how often such nodes are created, how many users (anon/auth) does your website have, which other modules you have installed, what's your workflow setup (e.g. Rules, moderation modules, etc.) and so on.

While doing this through your code is definitely going to be faster, I doubt you will really notice performance improvements. Doing things in code will in most cases have lower overhead than installing a module to do the same.
If you're really wondering about exact performance differences, you will have to benchmark it with Devel and xdebug for example.

However, you already mentioned another important point - it's generally not a good practice to have raw PHP code entered anywhere in admin. Sometimes it's faster to do it (e.g. in Rules, Views or other contrib modules), but over time you should look to migrate those changes to code. Not to mention that you will reduce the number of modules on your website, which is generally a good thing.