This is the new default password methodology used in WordPress 4.3.
In WordPress 4.3, if you have Javascript disabled or broken, then you will get the normal New Password and Repeat New Password fields. If you have Javascript enabled and running normally, then you only get the Generate Password button.
If you get two things on two different sites, then on one of them, you’re not running Javascript for whatever reason. Could be something in your browser blocking Javascript code.
If you see the Generate Password button, and click it, then it will generate a strong password for you. You may choose to override that password and type in your own. If you use a weak password, then a checkbox will appear to confirm your decision to use a weak password. Either way, it won’t stop you from using a weak password, and there will not be a Repeat Password field.
On the whole, the use of the Repeat Password field is actually less secure. The main reason for it existing was because passwords were typically hidden behind dots. You’re typing blind. Therefore, the repeat field is to make sure you typed it the same way twice. However, this encourages passwords that are easy to remember and/or type. Actually showing the user the password as they type it makes them more confident that they typed it correctly, and eliminates the need for the second field. Additionally, adding in live confirmation about whether the password they have chosen is a good or bad one encourages them to pick good ones. The checkbox to confirm that they want to use a weak password encourages them to not use a weak password in the first place, and to make a stronger one.
This new password flow is designed to improve password security and reduce password related problems.