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A New Balance 2E width is a 'true' 2E - i.e. not only is there more fabric in the upper giving it a greater circumference round the foot but the actual midsole is also slightly wider (about 3mm)
An Asics 2E just has more upper material but sits on the same width midsole as the D width.
I have found that an Asics 2E doesn't feel quite as wide as a NB 2E - but wide enough!
I disagree with the above comment. I have wide feet and find the Asics 2E wider than the NB 2E
I normally run in NB1064 but don't like the new 1080 so opted for the Asic Nimbus where I found the 2E slightly wider in feel. Also Asics appear to be more closer to shoe size than NB, so you may need 1/2 size smaller.
I think my suggestion would be to go to a running shop and try before buying, rather than internet shopping.