Э is more open and means that the previous consonant is always hard. The letter only occurs in the words of foreign origin to mark this hard pronunciation of the vowel. There are not so many words with this letter.
For example: это, дуэт, поэт, эхо, этика, эстетика
Е has 2 pronunciations:
1. In the beginning of the words and after Ъ/Ь it sounds with an й sound in front of it (as in yougurt or Yale):
ем, есть, платье, подъезд
2. After consonants it sounds as [е] or [и] (depends on the word stress), but has no й sound. It designates that the previous consonant is soft.
[е]: мел, тело (when stressed)
[и]: река, тепло (unstressed)
I think it's easier to listen to the words and compare the sounds.