The meaningｓ are the same. They are 99% interchangeable.
You can use both words in modern Japanese.
The difference is the pronunciation system.
一ヶ月 ＝ 一箇月 or 一個月.
Ikkagetsu = Ich/ka/getsu, which are all 音読み, Chinese-origin pronunciation
hitotsuki = hito/tsuki, which are all 訓読み, Japanese-origin pronunciation.
This is the same type of difference as that between 一個 and ひとつ
ひとつき sounds "softer" than 一ヶ月, as it always goes with other Japanese-origin pronunciation (訓読み）and that of Chinese origin(音読み）
You can use whichever you like. It is a matter of preference.