Now, of course, three phase power can be either wye or delta connected and
there is a difference between these two kinds of three phase power. I'm
sure you can explain the difference to us in simple terms and which kind is
used for general power distribution and why. When is the other kind used?

Yes, there is a difference, but it is pretty small ... one more
conductor. :-)

Actually there is quite a bit more difference:

Delta and Wye Power
In the United States, there are two main types, or topologies, of three
phase power. These are called Delta connected power, and Wye connected
power. There are some similarities between Delta connected power and Wye
connected power, and many differences. It is important to understand these
two varieties of three phase power in order to properly specify power for
your critical loads.

Delta Connected Power
Delta connected power is developed from three, independent transformer or
generator windings that are connected head to toe. There is no single point
common to all phases. Delta power is named after the schematic resemblance
of the windings to the Greek letter Delta.

A Delta Connected Source
Delta connected power is not commercially used in Europe. Three-phase
European equipment requires Wye-connected power.

Wye Connected Power
Wye connected power is developed from three, independent transformer or
generator windings that are connected at a common point, called a neutral
or star point. Wye power is named after the schematic resemblance of the
windings to the Greek letter Wye ( Y ).

Delta vs. Wye Power
Wye connected power has two different voltages available. The Phase to
Phase voltage is the main system voltage (typically 208 VAC or 480 VAC in
the United States). The Phase to Neutral voltage is also available, and is
typically used for small single phase loads (120 VAC or 277 VAC).

Delta connected power only has a single voltage level available: the Phase
to Phase voltages. Other voltages can be obtained only by using step-up or
step-down transformers.

Equipment designed to operate from Delta connected power, such as air
conditioners or motors, can also operate from Wye connected power without a
problem, since the Phase to Phase voltages are available in both systems.
However, equipment that requires Wye connected power cannot operate from a
Delta connected source. The Phase to Neutral voltages are not available. A
special isolation transformer, designed to convert Delta to Wye, is used in
this case.

Grounding of Delta and Wye Systems
It is common practice to ground the neutral, or center point of a Wye
connected source. In most cases, this grounding is required by the United
States' National Electric Code (NFPA-70). By grounding a Wye system, the
voltages to ground are stabilized and controlled. This makes a system much
less susceptible to impulses, ringing transients, and faults that cause
high voltages to ground. A Delta system is not required to be grounded,
although some Delta systems are grounded. Usually these are grounded to one
phase or to a center tap of one of the phases. This type of grounding is
rare, however. In most cases, Delta systems are not grounded.

Ungrounded or Floating Sources
Ungrounded sources are often called Floating, because they float with
reference to earth ground. System and wiring capacitance determines the
voltage on any system point with reference to ground. As a result of this,
the Phase-Ground voltages on an ungrounded source are very susceptible to
power quality disturbances.

The above is from http://www.teal.com/resources/an15.htm