Terminal services allows you to remotely run the program on a server where
keyboard and mouse input goes up the wire and screen updates come back down.
This is an advantage whenever the connection is slow enough that the usual i/o
the program requires is significantly more than the bytes required to do the
keyboard, mouse, and screen updates.  So, advantage one is performance (when the
aforementioned is true).

Another big advantage is that administrators only have to "administer" the
single or small number of servers rather than ALL of the client PCs that are
using the program.  Need  a newer version of an ODBC driver?  Install it on the
servers and your done.  If you're using a client/server set up you would have to
distribute and install that driver on every user's PC (and then hope it works).

In addition to a smaller number of machines to maintain, those machines can be
significantly more powerful than the PCs the remote users have so there can be
performance and cost savings from that aspect as well.

It's not a silver bullet and there are some downsides and hassles to deal with,
but it does solve a lot of problems if you need to run a program over a slower
connection.

Despite what many might think, using TS does not reduce software licensing
costs.  Every remote user is required to hold proper licenses for all software
that they run on the remote server in addition to a license to connect to the
server in the first place.