What have you been reading? Part of the problem is that both firearms and bows are classes of weapons, with many variations in each category. Any general observation is likely to be wrong for some significant fraction of weapons that were used by some culture somewhere - especially before the gunpowder synthesis of the mid-16th century resulted in relative consensus on the design and use of gunpowder weapons.

To break it down:

were very inaccurate

There's a tendency to impose standards on these weapons that people wouldn't do for bows, part of a presentist attitude that views firearms in terms of what they'd eventually become instead of what they were at the time. So without qualification as to what kind of accuracy, this is hard to answer. Arquebusiers were often mixed in with crossbowmen, if that's any assistance in figuring out how they were considered at the time.

were useful only at a very close range

As for short range - that was the culmination of a long period of tactical exploration that was in no way a monopoly outside of Europe. The famous Janissaries, with their long-barreled weapons and accuracy training, are one of the clearest examples of that.

they were actually effective

If used properly, without question. Still, that's probably true for almost every weapon system in its time - why else use it?

their killing range was generally equal to the killing range of bows

True enough, I'd refer to the earlier note about mixed units of firearm and crossbow equipped troops, and add that almost all firearm tactics evolved or were directly adopted from crossbow tactics.

had superior armour piercing capabilities at that distance.

Armor penetration dropped off rapidly as distance increased, due to the horrendous drag effects exerted on a bullet. I wouldn't be surprised if what you say was true, though the ballistic nature of the weapon encouraged close-range use against heavily armored opponents to maximize the chance of getting through (let alone the shock effect of a point blank volley).

Bit of a false dilemma, or at least not one that bothered the early 16th C. Spanish who landed in Mexico. They used both arquebuses and crossbows almost interchangeably, and often spoke about them in the same breath. Take this passage from the 1552 biography of Hernán Cortés (trans. Simpson:

Cortés reviewed his troops the second day after Christmas and counted 40 horse and 540 foot, 80 of whom were armed with crossbows or arquebuses, and nine guns, for which, however he had little powder. (p. 239)

Or this passage a little later:

[The Tlaxcalans] were very anxious to encounter the Mexicans, so Cortés sallied forth with them, and with 25 horse, 300 Spanish foot (including 50 arquebusiers and crossbowmen), and six guns. (pp. 251-2)

The reason for this may be that they were fairly equivalent, at the time, in a number of metrics such as range and rate of fire. Hassig (1994) shows off his penchant for quantitative data by calculating both of those metrics, given here:

• Crossbow - Range: 64m flat, 320 arc; ROF: 1 shot every min

• Arquebus - Range: 137m (only effective about half that); ROF: 1 shot every 1.5min

There is, of course, room to tease out the ramifications of the the differences between those numbers; talk about the ROF of higher draw crossbows; the "muzzle" velocities of each; various differences in upkeep and skill; questions of accuracy; and/or the psychological effect of either. In practical terms though, the crossbows and arquebuses at the time served in considerably overlapping roles on the battlefield.