British vs American English in their accent

In words which have the phonetic sound / ae/ , I hear the real differences between American and British Accent.

If in British they spell that phonetic sound similar to /a/, American would do with a sound like /e/, isn't it?

Some words such as slang, man etc. I wonder why they use the /ae/ sound for this? Do they mean, you can use whatever /a/ or /e/ when spelling the words?

Today I took one of the exam implemented in American English, I spoke the word: slang with an /a/ to the examiner, he didn't understand at all. When I said: "synonym is: vernacular language", he asked me:" Isn't it slang /e/ ?" . So confused!

Hi,

the symbol that looks like ae is actually one symbol, not two letters together, a + e. It's the IPA symbol used to represent a sound that is common in English.

Bet is pronounced with /e/.

Bat  is pronounced with /ae/. It's not /a/, and not /e/ either. It's between those two sounds in the IPA chart. To try to do this sound, you can try to say /e/ while your mouth is as open as for an /a/ sound. You move your mouth as if you were going to say /a/, but you try to say /e/ instead.

It's a symbol for phonetic transcriptions and it represent a sound, not a letter. It's not a matter of pronouncing it differently, but a matter of using it or not. Americans use that sound in "bath" for example, but some people in the UK don't. So Americans say /bæθ/, and some people in the UK use another sound instead, so that transcription doesn't work to describe the way those British people talk.
That said, the transcriptions you find in dictionaries are not perfect, not 100% accurate. For example, even if every dictionary says "can" is pronounced /kæn/, most Americans say either  /kɛən/, or /keən/, or /kɪən/. It often happens before nasal consonants like N and M, for example.