Julius and Augustus didn't add a month. They replaced a month. The 6th and 7th months had different names, and they simply renamed them to July and August. The actual reason why the month names don't match up is because Romans originally only counted "business months," kinda akin to how we count weekdays way more than we count weekends. In any case, no (or little) business was happening in the winter, so they just simply don't count months during the winter. Those uncounted months would correspond to January and February. When January and February were added, people decided to put them in the front of the calendar rather than at the back. Hence, the 8th month (October) became the 10th month, and so on for all the months.

You can read a bit more from the Wikipedia page here: https://en.m.wikipedia.org/wiki/Roman_calendar#Romulus