When comparing one treatment to another, why would you use the within
subjects design as opposed to between subjects design and vice versa,
what are your considerations?

Using a within-subjects design to test two treatments has the
advantage of letting you know that both treatment groups are identical
-- because they are one and the same group. This assumes, however,
that neither treatment causes a permanent change in the group. If it
did, then when the group starts the second treatment it is not the
"same" as when it started the first treatment, which is a big problem
when interpreting the effects of the second treatment.

With a between-subjects design, there are two different groups from
the outset, so there may be certain pre-existing differences that can
affect how the treatments work, making comparisons a little more
difficult. If you use random allocation to groups, making sure that
potentially important variables (sex, age, etc.) are balanced, then
these differences tend to be minor. This design would be useful if a
treatment actually causes a permanent change in the subjects.

So it depends on the particular treatments involved. I've interpreted
your question (through the use of the term "treatment") in a clinical
sense; for other types of experimentation, things may be different.

For example, I run cognitive psychology experiments (hence my Google
Answers nickname) where the use of within-subjects designs are
generally preferred because of the strictor control of extraneous
group variables. Also, all subjects are exposed to each of the
conditions (~ treatments), albeit in a randomized manner, so I can
compare across conditions without having to compare across groups,
where the latter can add more variation or "noise" to the results. On
the other hand, for certain types of social psychology experiments,
between-subjects designs are the only way to go because once the
subject is exposed to one level of a manipulated variable (e.g., happy
mood induction), it is difficult to expose them to the other level
(e.g., sad mood induction) because chances are the first exposure had
a significant effect that doesn't go away completely. So, like I said,
it depends on the experiment you want to run.

"strictor" should be "stricter"

Perhaps I should be stricter with my spell-checking :-)