How to pronounce the two and distinguish them in pronunciation as I almost hear them the same.
The s on the end can be difficult to hear, but both words are usually paired with a determiner:
I'll go in a month
I'll go in a few months
I'll go in one month
I'll go in two months
The word month always needs to be used with a determiner, such as "a month" or "one month". The word months is different because it doesn't always have to have a determiner:
This project is months away from being done. Correct!
This project is month away from being done. Wrong!
The "th" sound is one of the most difficult phonemes in English for non-native speakers, as many other languages don't have such a sound. It's quite possible that there is nothing wrong with your hearing, but rather your interlocutor isn't able to pronounce "months" correctly. In that case, rely on context cues as @Mark suggests. As with other countable nouns, "months" should be preceded by a determiner such as a, one, two, etc. If there is no article, then it's plural. The month(s) is trickier; you'll have to think.
I'll be away for a month(s).
"A" implies singular.
I'll be away for a few month(s).
"Few" implies plural.
It's that time of the month(s) again.
Only "month" makes sense.
I suffer from seasonal allergies during the month(s) of June and July.
Only "months" makes sense.
I suffer from seasonal allergies during the month(s) of June.
Only "month" makes sense (even if it repeats every year).
That's more than I want to pay for what ought to be just one month(s) worth of work.
Trick question. The answer here is "month's" — singular possessive.
Note that in practice, "months" should often sound more like [mʌnzs] than [mʌnθs]. That is, in fast speech, the /s/ sound should never disappear completely, but the /θ/ sound might not be distinct.
In the Midwestern American dialect, months is often pronounced "mons" (i.e. məns or monce), emphasizing the "s" sound and virtually ignoring the "th" sound.
But honestly it virtually never matters. If the number of months is important, then it will be specified with another word indicating the number (e.g. "a", "single", "one", "two", "few", "many"). Using month instead of months may identify you as a non-native speaker, but it should never cause any confusion.
As a native speaker, I frequently drop the plural marker from words simply because they seem pointless. E.g. I might say "bring me two beer" or "I saw some few turtle". That is a rather idiosyncratic trait of mine, but people never seem the least bit confused by it in spoken communication.
Similarly, people sometimes use a plural marker erroneously. For example, one might say "do we have pens?", when they only need one pen. Or they might say "I have siblings" when they only have one sibling. This is probably due to the fact that people tend to use the plural form when the number is unknown. Also it takes fewer syllables to say "I have siblings" than "I have a sibling".
To distinguish the two words when listening, listen for the ss hissing sound at the end of "months". It's not a z, as many final esses are, it's a short hiss.
To pronounce the word, if you're not accustomed to pronouncing a theta, you can substitute "t" and add the small hiss for the plural. It'll be easier, and many native speakers pronounce it that way: "muntss".
Then, when you can easily pronounce that form, you can stick the tip of your tongue between your nearly closed teeth and try out the theta, followed by a quick pulling back of your tongue to sound the hiss.