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
ひと月
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
東京に来て一ヶ月が過ぎた。
東京に来てひと月が過ぎた。
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.