They mean different sounds. Е stands for /e/ in "red", Є stands for /je/ in "yes".
When after a consonant, Є means the palatalization of that consonant ( the "y" sound is lost in that case).
Є after a consonants mostly happens when we try to write Russian words with Ukrainian letters :D (couldn't actually think of Ukr. word with this combination)
There are really few words with such combination, one of an example can be 'ллється', which means 'is pouring'
You mean, if you were to transcribe the same words in Russian е →э, є→е ?
This is SORT OF true. However, Ukranian "е" is a more open sound than a typical Russian "э". Also, etymologycally speaking, many Russian words with "е" have a corrresponding Ukranian words with "і" or "ї".