I think there is a very subtle difference, but nothing that wouldn't make you unable to be understood. The "Ж" sound is a little further back in the throat, and the "Щ" is more similar to the "sh" sound that is used in English. Though, I could be wrong, as I am still learning.
When a word is made plural in Russian, the last letter is "и"or "ы" depending on whether the last letter is soft or hard vowel or consonant. Russian words that end with a consonant is added "ы" to make it plural, if a Russian word ends with А,Э,Ы,О,У vowels, it is taken off and swapped with "ы"
If a Russian word ends with the vowels Я,Е,Ё,Ю,Ь,. You take the vowel off and add "И" instead when making it plural
In this case, both Мужчина (man) and Женщина (woman) ends with "а" and is therefore, hard.. so we take off the hard vowel and replace it with "ы" when made plural. Мужчины (men) and Женщины (women)
Some other exceptions such as the 7 letter spelling rule apply, you can search it up for more help.