Blast results are not symmetrical, meaning that blasting A to B will not necessarily produce the same result (score etc) as blasting B to A . This is a common misconception. One of the (most important) factors is indeed, as you mentioned, the blastDB size (you can 'resolve' that partially by using a fix DB size, cfr the blast manual for the syntax) and will mainly influence the E-vlaues score. The bitscore is more directly related to the actual alignment and will thus not differ (if at all) as much as the e-value.