In Russian, pronunciation of Е depends on its placement. If preceded by a consonant, it's pronounced like a "softer" Э. A bit like "bet" vs "bat" in US English. It's only pronounced as "ye" if it's the first letter of the word, or preceded by a vowel, soft sign (Ь) or hard sign (Ъ).

In Ukrainian, Є is similar to Russian Е, but it's not modified if preceded by a consonant - in these cases it's typically replaced by i. Ukrainian Е is indeed about the same as Russian Э. And Russian Ё is replaced by combinations ЙО or ЬО.