Livingstone is the little town from which visitors to Zambia can visit the Victoria Falls, relax on the Zambezi or indulge in the multitude of crazy adrenalin activities available. Unlike Victoria Falls town in Zimbabwea which is within easy walking distance of the Falls, Livingstone is about 6km from the Falls itself and you need to get in a car to get there. Whether the Falls is best viewed from Zambia or Zimbabwe is a rather subjective issue and depends on who you talk to and the time of year. From Zambia, you look along the gorge and the area of the Falls immediately in front of the Zambian view points can dry up at certain times of the year. Both countries suffer when the river is in spate as you see an awful lot of spray and not much of the falls themselves...you also get a free shower.
One advantage of basing yourself in Zambia is that there are a number of camps in great locations which are worth visiting in themselves and offer a number of activities in the park or on the Zambezi. Most of the camps are actually on the river and are between a 20 and 45 minute drive from the Falls. You can generally spend longer (ideally 3 nights) in these camps enjoying the environment and relaxing when you are not doing the rafing, bungi or other Falls activities. Of course, it is less easy to be independent here as you need to get transfers to and from the Falls but the camps are well-equipped to organise these and your activities for you.
Livingstone itself is quite a small town but larger than Victoria Falls. It has a number of nice coffee shops and a few restaurants. There is a decent craft market to visit and a couple of good museums if that is your thing. It has a pleasant, unhurried, African-time air about it.