I love this board! A question about why wear a watch becomes a debate on whether a Rolex will implode at 800' as implied by some now dead guy - beautiful (kidding aside, Sheck was the MAN in the cave diving world - RIP).
A dive watch is a necessity if you don't have a dive computer. As has been pointed out, it provides one of the three critical pieces of information needed to safely stay within NDL limits (regardless of which tables you dive). A dedicated bottom timer can also provide this information, but a dive watch set correctly at the beginning of a dive is just as effective and has uses outside of diving. I wear a watch while diving as a backup. If my DC were to die, I can safely ascend to my safety stop and use my watch to time it.
A dive computer is a step up from this and is considered a "must" by many divers and even by some dive shops/operators for multi-day, repetitive diving. After a recent trip to Coz where one new couple chose to dive without even a watch - "we'll just get in last and go up first" - our shop owner/DM vowed that he would not take anyone without a computer again. Maybe a bit of an over-reaction, but you get the idea. DC = the basic info you need to stay within NDL, and is a solid safety investment if you have decided to actively pursue diving. OTOH, if you are going to dive once or twice a year at relatively shallow depths (<60') a DC is overkill IMO.
Whether to get an air integrated (AI) DC is the next choice. I dive an Aeris Elite T3 hoseless wrist mount computer. When I decided to buy, I wanted something that would grow with me as I advanced in diving. It is Nitrox capable, has a gauge mode, supports 3 different tank/gas mixes, plus I liked that it was a wrist mount as I never have to "look" for my gauges ;-)
Hope this helps.