Bonus armor penetration was an attempt to reduce the potency of assassins by only affecting armor gained from runes, spells, effects and armor, this was considered bonus armor, while every champion has armor specific to the unit, called base armor. Bonus armor did just that, penetrated bonus armor, so say someone has base armor of 100 and bonus armor of 300, the bonus armor pen would only affect that 300 value, while total armor pen (or simply armor pen, but I will user total armor pen for ease of understanding)) affects the whole 400 value.
However, with the changes intended to ship, armor pen is likely to be stronger than before as the pen value is higher.
Another example is a unit such as an adc, with 80 armor and no armor items, bonus armor effects would have no effect what-so-ever to their defense, thus they have the full benefit of 80 armor (which is 44.4% damage reduction), but with total armor pen, this defence value is affected, reducing the protection to 34.2%. I shall assume you do not understand how armor pen works either, as it is likely you would simply assume that it removes 35% of the reduction you have, or 35% of the reduction you already have (which would be 9.4 and 28.86 respectively, however, this is supposed to be incorrect), the armor value is initially reduced by the penetration % (35% or 25% depending on the item) so the 80 armor, when reduced by 35%, becomes 52 (0.65*80=52), which is then applied to the formula (100/(100+armor), or simplifying the equation to include armor pen by using (100/(100+(armor−(1−armor × armor pen)))), which results in the same final answer. As such total armor pen affects all enemy champion units, whereas the bonus armor pen only affected units that have bonus armor from spells, effects, items, and runes.
Hope the extra info didnt hurt
TDLR: Bonus armor pen is weaker, total armor pen effects the full armor value, bonus armor pen does not.