How does the full-size board change the game of Go?
The small board focuses on close fighting. You're right; it is recommended for beginners for developing fighting skill and tactics, but the strategy component is lacking--at least when compared to a 19x19 board.

The big board is not just like playing 4 small board games next to each other. Through ladders and potential escape routes, the stones on one side of the board interact with the stones on the other side. When you start playing on big boards this is a very hard part of the transition because on a 9x9 or even 13x13 board you don't have to keep thinking about stones that seem to be far away. This is related to the concept of "influence", which is largely absent on smaller boards, but is of great strategic importance on the bigger board.

A Go teacher in my area likes to say that what makes professional Go games really interesting is the trades. You see professionals let massive groups die yet come out ahead, because in so doing they gain forcing moves that make up for it elsewhere. A popular example is the Yi Sedol ladder game. These sorts of trades are nearly impossible on a small board, and the game is much more exciting with them.

Overall, on a big board Go


is more complex, which many would argue means more interesting
uses both tactics and strategy
for an amateur, allows for more mistakes. You can mess up in one area, but still come back, playing better elsewhere and using leftover aji
allows for more varied games. There are many different opening styles on a full size board, and they will lead to games that feel very different.
is more fun!