Its possibly easier to make comparisons TA vs SA rather than pros and cons.
In general TA's tend to be larger and heavier caravans (There are some exceptions to that generalisation) They also tend to be more expensive to purchase.
Consequently you are likely to need a bigger tow vehicle along with their associated running costs.
Most dealers charge more to service a TA, than an SA.
You have four tyres to change when they time expire.
TA's are very difficult to manhandle around bends, because the you will always be trying to scrub two tyres partly sideways.
Setting the nose weight is quite critical and needs great care.
Leveling a TA on site is a little more troublesome than an SA.
None of these points are not insurmountable, and with a little extra care and planning most TA caravanners on average enjoy the benefits of a slightly larger caravan.
But As I stated above TA's are not always the biggest, there are some large SA's
There is a debate about towing TA's and inherent stability afforded by a second axle, I believe TA's do tow with less apparent disturbance from other vehicles, but that just my experience.
Having used both TA's and SA's, I can honestly say I have no overriding preference for either of the configurations.