I don't know how "" is used, or who uses it. Usually slashes "//" represent an IPA pronunciation. For example the WR dictionary says "dress /drɛs/" where they are using IPA symbols, but uses "dress (dres)" in the second part, where they don't use IPA symbols.
Dictionaries often use "//" for pronunciations, but may not follow the IPA rules exactly. The article you linked points this out for /ɛ/ and /e/ in different dictionaries. Another common mis-use is /r/. The IPA symbol for the Engish R sound is /ɹ/ but many English dictionaries use /r/ instead.
To listen to the exact IPA souunds, try the chart at this website. Just click on any vowel or consonant to make it play the sound. Clearly /e/ and /ɛ/ are different:
The square brackets indicate pure IPA sounds: if you know the IPA, you can pronounce a word approximately correctly without knowing anything more about the language. (This is not wholly true: they might contain more or less detail; a less detailed notation might be less accurate.)
The slashes are more specific to one language. The notation /dres/ tells you that 'dress' has the same vowel /e/ as in 'set' /set/ and 'said' /sed/ and 'any' /eni/. This is true in all accents (or almost all). However, accents might differ in exactly which vowel they use here: most accents use the vowel [ɛ], but some might have something closer to [ɪ] or [æ], and it might vary depending on the surroundings. For example, some accents merge 'pen' and 'pin', so for them, /e/ is pronounced [ɪ] before /n/. In most accents 'said' is longer than 'set', so /sed/ is [sɛˑd] (with a 'half-long' symbol [ˑ] after the vowel).
So why would we ever use /slashes/ if [square brackets] are more accurate phonetically? Often people don't need phonetic detail. They just want to know that 'said' is pronounced /sed/ with a vowel like 'bed', not /seɪd/ like 'made'. This is what a dictionary needs to show.