Yes, electric toothbrushes do a much better job, and every dentist I've gone to in the past 10 years or so has recommended them for that reason. Let's face it, they move much faster than the human hand can move a manual toothbrush, therefore they get the teeth much cleaner. Also, many of them are pressure-sensitive to prevent you from brushing too hard and damaging your gums.
Many dentists sell electric toothbrushes for much less than what you would pay in the store, so call your dentist's office, ask which brands/models they sell, and ask the price.
I use a Braun/Oral-B Professional Care because that's what my dentist recommends. He used to sell the Sonicare as well, but he said he stopped carrying it because his patients didn't like it. Apparently, many people find it uncomfortable to use.
I haven't tried the $10 battery operated variety, so I can't comment on how well they work compared to the more expensive models. My guess is that even if they work as well (or nearly as well) that they break down sooner and need to be replaced more often.