In electricity, there is a thing called amperage capacity, and
that term is sometimes shortened to "ampacity". Basically, it is
a matter of making room for electricity to flow, without stuff
getting too hot.
Think about the electrical wiring in your house for a moment.
The kettle and toaster, have thicker wires on them, than the
lamp in your study. Thicker wire is used to carry heavier currents,
without the wire becoming too hot.
There are two ways to make a thicker wire. The obvious way,
is to pack more strands into one cable (duh). Another way
(which is not used for house wiring), is to run two wires from
the same source to the same destination. That allows more current
to be carried safely.
Your ATX power supply has several kinds of amperage limits. One
limit, is the connector pins used. There are limits as to how
much current each pin can carry. The wires can generally take
more amps than the pins can, so the pins are a limit. You don't
want the pins to get hot, because they can melt the plastic.
The ATX main power connector already has multiple +5 and +3.3V
pins and wires. There was a time, when PC designers thought even
more +5 and +3.3V would be drawn, so they asked the power supply
companies to add a six pin AUX connector. A motherboard designer
would only add that 1x6 connector to the motherboard, if it was
felt that room for the additional current was necessary. (Connectors
cost money, so they don't slap them down for no reason.)
The era of needing extra +5 and +3.3V is past. So the Aux
connector has been removed from a lot of new supplies, and you
won't likely see it on a desktop motherboard.
The +12V 2x2 square connector has a similar history. At one
time, the main connector had enough room for the current on
the +12V. Now that +12V is used to power the conversion circuit
that feeds the processor, additional current is required. A
130W P4 processor would need more than 10 amps from +12V, so
they needed more wire to carry the current. That led to the
introduction of the 2x2 12V connector. For the majority of
P4 or Athlon64 motherboards today, that connector is mandatory.
So when you find a connector on a motherboard, it is there for
a reason. Best to connect it, if you've got it