What is the difference between בת זוג and אשת(the wife of)...?
The possessive form of woman like, אשתי, meaning my wife
ESHET is a little archaic (differently from ISHTI). See more below from Jessika and me.
I think part of the difference is that the older forms of husband/wife were wife - אישה which just means woman, and husband בעל which means owner. In the modern age calling your wife woman is rude. And your husband being your owner... yeesh.
The more modern בת/בן זוג means partner or spouse and is less troublesome from the PC point of view. As far as the difference between בת זוג שלי and אשתי goes is that the former is more modern and is easier to use with same sex couples than the traditional, second form.
I'm not a native speaker, so don't take what I say as writ, but from what I've seen in and heard in regards to this it's pretty straightforward and usually interchangeable, at least with younger people.
Excellent explanation. They are interchangeable in every day use. Most Israelis for example would use ISHTI (my wife), which is still different from saying HA-ISHA SHELI (my woman), although it literally means the same thing. People who are particularly aware of PC/language bias, would use BAT-HA-ZUG SHELI (my partner/spouse) or more colloquially ZUGATI (my partner/significant other). BTW, ZUG means COUPLE, so BAT-ZUG is somewhat similar to MY OTHER HALF.
Thank you! It's nice to hear that I've got a good grasp on a concept! I've not been studying long enough to be confident in most of my Hebrew studies.
i like זוגתי myself, completely interchangeable :D And we use 'my other half' as a kind of affectionate form of address in the States as well.
 The woman that you are married to is called אישה = wife [i-sha]; my wife = אשתי [ish-ti]. The man that you are married to is called בעל = husband [ba-'al]; my husband = בעלי [ba-'a-li].
 Couple is זוג [zug] and it has two meanings (thank you google translate!): -1: two individuals of the same sort considered together -2: two people who are married, engaged, or otherwise closely associated romantically or sexually.
A female spouse/partner = בת זוג. A male spouse/partner = בן זוג.
 The expression בנות זוג [bnot zug] is used if the couple is formed from two women. "אנחנו בנות זוג" = We are a couple (of two females). The expression בני זוג [bney zug] is used if the couple is formed from two men.
 Some people prefer to say זוגתי [zugati] = my female partner. It has exactly the same meaning. The form זוגי [zugi] = my male partner is not really used.
 In formal occasions and speeches some people may used the form: My wife = רעייתי [ra-'a-ya-ti], from רַעְיָה [ra-'a-ya], which means "my beloved [female] one", and in modern Hebrew could also means wife. For example: ראש הממשלה ורעייתו [rosh ha-mem-sha-la ve-ra-'a-ya-to] = the prime minister and his wife.
The word "רעי" [re-'i] = "my beloved [male] friend" means a dear friend, and even in formal speeches and poetry it's rarely used in the meaning of husband. An alternative formal variation for my husband is "אישי" [i-shi], literally = "my man".
 Some people don't like the word בעלי - The word בעל also means owner, so בעלי could also be interpreted as "he owns me". Therefore they may prefer the form that was mentioned earlier בן זוגי, even if they are married. Others will use the formal variation "אישי" - but if you'll do so, people will ask you for the reason behind it.
There's a detailed discussion about this in the Academy of the Hebrew Language - (obviously, it's in Hebrew): http://hebrew-academy.org.il/2011/05/01/בַּעַל-האם-ראוי-למצוא-תחליף-למילה/