One difference may be as follows: When someone is extremely
limited with resources, and on the second night of Chanukah
lights only one candle, if on the third night he has
sufficient resources, how many should he kindle?
According to those who assert that Beit Hillel's opinion is
based on the days that passed, he would have to light three
candles. According to the explanation that it is necessary to
increase in holiness and not decrease, it would be sufficient
to light only two candles, since the night before he only lit
one.
Another difference would be in the event that on the third
night of Chanukah one had resources sufficient only for two
candles, how many should he light? If Beit Hillel's opinion
follows the days that passed, in this case, since one cannot
light three candles, one will light only a single candle, in
order to fulfill the basic obligation of lighting a candle
each night of Chanukah.
According to the other explanation, one would light two
candles. Though one cannot accomplish the increase in matters
of holiness, at least one will not violate the command of
not to decrease in matters of holiness.