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I just did some dives with my newly cert. daughter who trained on dive tables. When she saw how much more time was gained diving with a computer vs. the tables all she could think about was getting her own dive computer.
A big thing about dive computers is they give you more flexibility in executing your dive plan. Maybe you've planned your dive to a max depth of 50 feet, based on what was explained in the dive briefing, but then when you get down there, you notice that there's a really cool critter hanging out at 65 or 70 feet. With a dive computer, you have the information at your fingertips to decide whether you can go get a closer look, but if you have just a watch, unless you carry tables on the dive with you, it's more difficult to make that judgement.
I realise that a dive compy can tell you everything from the temperature, to the rate of ascent, to your arsehole tightness, but for someone starting out, would it be doable with a watch?
I would recommend a watch and depth gauge over a computer. Not just to save money but to reduce your dependency on something that can quit on you if the battery fails. The other problem with computers is while they can give you more bottom time it is at the expense of reducing your safety margins. To avoid carrying tables with me I used permanent ink to write the no decompression limits on the arm of my wet suit
a dive computer will give you deco information if you accidently went there.
you would need a lot of arms or your wetsuit to sort that lot out.