Why do some appliances tout watts and some others advertise volts. I know watts is a measurement of volts*current but looking at just watts alone, what does that tell us about the appliance? Can't volts be low and current be high to get the same amount of power in watts?
Why is the watts measurement necessary is what I'm really trying to ask. Why not just provide volts and current?
There are two principal AC mains household voltages in use in the world today, (120 and 240 volts) and the appliance's voltage rating is given in order to indicate what input voltage the appliance is rated to support.
Watts is specified since that's the work that's being done by the appliance when it's ON, and that's what your power company is charging you for.
EDIT/UPDATE:
In the examples you've shown, the juicer ad is non-deceptive in that it discloses how much power it uses and, knowing that and what your power company charges for energy, you can determine its run cost.
For example, if your power company charges 10 cents per kilowatt hour of energy delivered to your home and your juicer uses 1 kilowatt, (1000 watts) then for every hour it's ON it'll use one kilowatt-hour of energy and you'll be charged 10 cents for that. Not a bad deal, and to look at it another way, if it takes you six minutes to juice some fruit and your juicer is running balls-to-the-wall, that'll cost you a penny.
On the other hand, there's nothing to be gleaned from the Sears ad since all that's mentioned is that the battery is an 18 volt NiCd pack, and that it has a capacity of 1.1 ampere-hours, which means nothing if the running current, under load, isn't specified.
It isn't, and even though the magical power number could be backed into knowing torque and RPM, the torque is specified, at 1700 RPM, as zero inch-pounds, shutting all the detective work down.
There may be more exhaustive data at DeWalt's site re. the performance of their drill.
Watts are specified when the device's power usage or power output is important to you. For example, two 2000 W hair dryer will produce the same heat, even though one might be American and run from 120 V and the other European and run from 240 V. The same is true of a microwave oven, a light bulb, a toaster, and a motor. Motor are often rated in horsepower, but that just a different unit of power. 1 horsepower = 746 Watts.
Some appliances nowadays have a universal power input. That means they can run from around 90 to 260 V and 50 to 60 Hz, which makes them compatible with normal line power anywhere in the world. The amps varies inversely with the volts, but the watts remain mostly constant.
Watts*hour is what you are paying for to the electricity supplier. It is a power, or energy per unit of time consumed by the appliance. Given your device is plugged into 110VAC or 220VAC wall outlet, you can derive the current consumed and estimate whether the outlet/entire home wiring is capable of powering it. If you see the voltage rating on the appliance, it is likely to indicate whether or not it will work with 110V or 220V. If some other voltages are stated, it is most likely DC voltages, for operating with some kind of AC/DC adapter or even battery, and the purpose of it to prevent you from plugging the wrong adapter in.
It's true that you can calculate wattage from voltage and current - the wattage given is redundant. For example, if I see a 20 V, 4.5 A laptop charger, I know that it's a 90 W charger, no matter whether it says that on the box.
The main reason why manufacturers give wattages is because not everybody knows this. Some people have a concept of how much power 90 W is and how much current 4.5 A is, but don't know how to calculate one from the other. Labeling products with both makes it easier for the average joe to figure out how much power this thing needs.
A very practical explanation for the outlier (the drill) is because the drill uses a battery pack (see the black handle in the bottom of the picture in your original link).
The juicer runs off of mains (AC), so (for the USA) the voltage is assumed to be 120 VAC, so wattage is specified such that current can be determined if you will overload the breaker for that line.
Voltage and Amperage markings are required per safety labeling (on the appliance). However for advertising proposes just about anything can be used to attract the customer. For eg: You may want to conserve power so you buy the lowest wattage device, or you want the strongest car battery so you look for the highest cold cranking Amp rating.
There are a lot of devices, where watts is the output, and voltage and amps is the input. For example my microwave states it is a 900W one, but it says that it draws a max of 6A. This can be especially true for all kinds of power supplies where the DC wattage is specified, but the AC power draw is higher, due to losses.
Also on quite some devices, wattage is a pure marketing number, giving some kind of overall power, whereas the momentary amps it draws can lead to much higher values for short times. This can be good to know when you don't want to trip your breaker...
