glad you asked.
UA had the 763 in its fleet before the merger. The 764s came from CO. The most common aircraft in UA's TATL fleet is the 757, most of which came from CO.
UA operates on average 26 flights/day with its 764 fleet. DL operates 38 flights/day on average with a similarly sized fleet but only 12% of its US-Europe flights are on 767s. 25% are on 777s and 5% are on 744s.
specific to AA, PMAA operates 43% of its flights from the US to Europe on 767s, the same percentage on the two types of 777s and the remainder on 757s. including PMUS, the AA/US TATL operation is the same amount of 767s but now a percentage of 27% followed by 332s with 18%. The 757 operates 17% and the 333 operates 13% of TATL flights.
In contrast, DL's 333s and 764s operate 46% of US-Europe flights and the 763 is 38%.
so the real difference between AA and DL on the TATL in terms of fleet is that DL uses 333s and 764s where AA uses 777s.
For those that accept that a 290 seat 333 is a more efficient aircraft than a similarly configured 777 (and AA and UA's 777s don't even have that many seats), DL has the most fuel efficient US-Europe fleet.
on the Pacific where the range of the 777 is needed, the 777's extra fuel burn is justified and given that AA doesn't have 744s that DL and UA have and AA is aggressively replacing 777s with 787s, AA probably does have the most fuel efficient Pacific fleet.
so, yes, fuel efficiency does matter.