It depends on the power requirements of the computer. Older segment based LCD displays use very little power, so watch type batteries can work well and last a long time. I'm one of the designers of the Atomic Cobalt:
Cobalt Guide: Home
And we use rechargeable batteries, as the power requirements of the color OLED display are higher than coin cells could supply, and we also have a bit more processing going on, which takes power. We get about 40-60 hours of actual display-on diving time on a fully charged battery, or about six months of standby power. Recharging from a full discharge takes about two hours, and can be done through a wall charger or through USB, including solar or auto chargers. It's nowhere near to requiring a charge every dive day- closer to every dive week- though there are some computers using color TFT displays that probably do use batteries that fast.
The internal discharge on modern Li-Ion batteries is quite low, but the computer "wakes up" to sample ambient air pressure every few minutes, and that takes some power. All batteries, including rechargeable, wear out eventually and will need to be replaced, for Li-Ion that is typically about 5-700 charge/ discharge cycles, or many years of diving.
One plus is that rechargeable batteries are a bit more eco friendly, as they have fewer toxics and are not thrown away as often. Also, in our case anyway, you can recharge without opening up the case of the computer. On the negative side, rechargeable batteries will be more expensive initially, as the batteries themselves cost more and, if they can be charged in situ, there needs to be charging circuitry in the computer.
IF the computer and the power supply are appropriately matched, there is no inherent advantage to one or the other, it's weighing ease of use and availability vs. cost. I expect more and more dive computers will come with rechargeable batteries as technology improves and they become more sophisticated, just as has been the case with other electronics.
Ron