This isn't about convenience. If your dishwasher can do the job without any rinsing or second cleaning, than it will save you some time but not money or energy
First off there is no contest if you don't already own an efficient model. The cost to manufacture, transport, and install a new machine will never be recovered. The machine will need service or have to be replaced long before any energy savings could be realized. Dishwashers use electricity to operate and gallons of burning hot water to do their job. I don't know about you, but I don't fill up my sink to wash dishes. I wash them, rinse them, and I can do a whole sink load with only a couple of gallons of water and a few drops of liquid detergent. I don't let the faucet run the whole time and I don't need hot water because I do the cleaning, not 140 degree water shooting out of high pressure nozzles. My sink will never break down or need repairs. I have replaced the water valve in the faucet body one time in 30 years, with a free unit from Delta Faucet who guaranteed the faucet for life.
If you already have a dishwasher installed you can at least try to compare the efficiency of the two methods of cleaning those dishes. As anyone has learned from using a dishwasher, the ability of a machine to do a good job of cleaning is based on how dirty those dishes are to start with. It also matters how well that unit works. Just looking over the reviews from buyers at www.sears.com reveals how poorly the cheaper models perform. A hand washer scrubs that one dish till it's clean because that dish is being visually inspected during the whole process. A machine just assumes that enough hot water squirting long enough will have done the job.
The efficiency of these methods really comes down to how much water is needed since the keyword here is how "hot" that water is. A dishwasher needs water at 140 degrees Fahrenheit to do the job as designed. So the first thing you need to do is turn your water heater up because few people have their heaters set this high. Of course having this water set this high also causes the 300 emergency room visits each day by children burned by household water. Turning up the water heater is also going to use more energy which isn't necessary when washing by hand.
Some models have heat boosting modes. This allows you to set your water heater back down to a safer temperature of 120 degrees. But the water heater doesn't care where the water is going. Once water leaves the tank the heater kicks on and starts heating up that incoming water. Add to that the energy used by the dishwasher to heat the water up to 140 degrees using its booster and the savings isn't going to be very much and it isn't using any less water. But this brings up some interesting things about a dishwasher. If it is so efficient why do I have to select all the most efficient options to make it efficient? If I select all these efficient modes will the dried on egg yolk come off? One of these options is the heat dry mode. Don't use it if you want to save some energy but it still uses electricity to blow air.
Then there is the claim that the detergent used in a dishwasher is also cheaper; ABSURD! A few drops of any dish washing liquid on a sponge will wash an entire sink of dishes. A bottle of dish washing liquid from Walgreen's can cost only $1.50 for a 13 .oz bottle while an 8 oz. packet of Cascade for a dishwasher also selling at a Walgreen's is $4.00 and that's for 12 wash cycles. You do the math.
If you really want to be efficient you will simply wash your dishes by hand and do it right after each meal if time allows. You don't need to wait for a full load like a dish washing machine, you won't make nearly as much noise, and as long as you pay attention you won't be inspecting any dishes to see if they got cleaned right the first time. If you have a machine save it for that party when a dozen people come over and you've got that pile to do but not the time.