Vow and promise here are the same things. It comes down to one’s intentions at the time of making such oath/vow/promise as Quran makes a person accountable for oaths taken with solemn intention as mentioned in Quran 5:89 – it is the presence of a solemn intention which makes it a binding covenant with Allah. Usage of words may vary from individual to individual ranging from swearing oath on Allah’s name, making a promise with Allah, nadhr (vow) etc. can all be considered as oath with Allah if done with a solemn intention. If there was no solemn intention, then merely using such words promise or vow do no make one accountable. Was your intention actually of swearing an Oath in front of Allah or not? if yes, then there will be kaffara as well.
If, owing to some reason, a person breaks or is forced to break a solemn oath, then he must atone for it. For this, he is required to feed ten poor people with the standard of food he normally feeds his own family or to give them clothes to wear or to liberate a slave. If he is unable to do either of these, he must fast for three days.
For details on Islamic Sharia on Oaths and their Atonement (Qasam Aur Kaffara-e-Qasam), please refer to the following thread:
Discussion 62809