AV asks the voter only to identify all the candidates that he or she supports. And while BAV asks this as well, it also asks the voter to identify which candidates the voter opposes. A common mistake is to identify non-support and opposition (no doubt a mistake rooted in the assumption that there are only two candidates). BAV avoids that mistake; when a voter marks neither support nor opposition (or perhaps by mistake, marks both) that vote is correctly treated by BAV as indifference to the candidate; it neither helps nor harms the candidate's prospects of winning election. AV (along with many other voting systems) have no way to distinguish indifference from opposition. While that may not matter when there are only two candidates, it is a misleading oversight in more contested elections.
With very little extra effort on the part of voters or election administrators, BAV provides accurate and quite important additional information about voter opinion in comparison with AV. As a bonus, voting with BAV becomes easier than with AV, if only by BAV providing a convenient way to express indifference to a candidate. In addition, being allowed to explicitly express opposition to a candidate is sure to be less frustrating for voters than being limited to somehow hinting about opposition.
And yet another reason to prefer BAV over AV is that BV corrects for an unjustifiable advantage that AV provides to famous candidates. Nearly everyone has an opinion when it comes to a famous candidate so famous candidates will suffer exceptionally few abstentions. In an AV election, that makes it only possible for fame to bring a bonus of extra votes of support. Even a notorious candidate will likely be rewarded by AV for the fame that notoriety brings.
Likewise, in a BAV election, fame, and even notoriety will likely bring in extra votes of support. But with BAV, that extra support can be accompanied by a compensating increase in opposition votes. So, in a BAV election, notoriety is quite likely to result in a smaller vote tally.
Last (but by no means least), is that unlike the other three other voting systems, only BAV provides a clear promise of ending duopoly. As polarization grows, the lopsided power of the duopoly fades and the opportunity for smaller parties improves, bringing competition into politics. BAV will make it more difficult to maintain a duopoly.
As a bonus, BAV will likely encourage a more moderate tone both in elections and in politics generally. Candidates will avoid taking extreme positions that antagonize voters because those voters can respond by turning abstentions into votes of opposition. And, as more parties come into viability, there will no longer be one single opponent to target for an attack.
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