Quick answer
Э = e as in “pet”, with no y sound. Е = ye. That is the whole difference, and it matters at the start of words.
Ээ
Ээ
e
э́тоthis
How to say it
Say “echo”. The first sound is Э. Now say “yes” — that first sound is Е.
э́тоthis
эта́жfloor
экза́менexam
по́этpoet
эне́ргияenergy
э́хоecho
Mostly in borrowed words
Э is rare in native Russian words and common in borrowings: эконо́мика, электри́чество, эмо́ция. If a word looks international and starts with an e sound, Э is a safe bet.
Э or Е after a consonant
After consonants Russian usually writes Е even when the pronunciation is hard: те́ннис is said “tennis”, not “tyennis”. That is a feature of borrowed words, and a dictionary is the only way to be sure.
Typical mistakes
Reading э́то as “yeto”
“eto”
Э has no y sound at all.
Writing эсли
е́сли
Native words take Е even when they sound close to Э.
Check yourself
Which word contains Э?
э́то — this.