I have not exactly the same problem but when entering the captcha code the application say that it is invalid, then it presents a new code and then it works.
I found out that it was because of the plugin WP-Rocket cache.
Contact Form version 4.0.3 with Really Simple CAPTCHA Version 1.8 is working for me on many different websites and several different servers. Any issue you have is almost certainly due to something on your local WP install.
Step by step approach to problem solving
1. Switch temporarily to a WordPress default theme (Twenty Eleven etc.) and see if the problem goes away.
If it does, you’ve most likely got a problem with your current WordPress theme.
If switching to default theme doesn’t help it could be one of the plugins you are using that is causing the problem.
2. Disable all plugins other than Contact Form 7
If doing that solves the problem, you can then add back each item – one by one, until you find the cause of the problem.
Definitively my problem is due to an error on WP Rocket cache plugin. Confirmed by the contractor. Will be corrected tomorrow.