There is some very good evidence that replacing a manual toothbrush with an electric toothbrush can help in reducing plaque and the risk of gingivitis.
The Cochrane Oral Health Group published a review1 that summarized over 50 studies from 1964-2011. A majority of the studies tested the effects of a rotating oscillating electric toothbrush against a manual toothbrush. Overall, they found a 21% reduction in plaque and an 11% reduction in gingivitis.
There are some things that weren't found by the studies that were reviewed, mainly what the long-term dental benefits of electric toothbrushes. Also, side effects were rarely reported, so there was no sufficient evidence of whether electric toothbrushes can be harmful to your teeth.
A study done in 20082 tested oscillating powered toothbrushes against manual toothbrushes over a 9 month period. It started with a 3 week period of oral care. Then some people brushed twice daily with either an oscillating powered toothbrush, a manual toothbrush, and a manual toothbrush and flood together. The powered toothbrush was able to keep the levels of plaque lower than the other two groups and also help prevent gingival bleeding. Another 2010 study3 also found very similar results in which the oscillating toothbrush help reduce plaque and prevent gum bleeding. From these studies and many others, we can see that oscillating toothbrushes are better for your teeth than manual toothbrushes.
There is another type of electric toothbrush, sonic toothbrushes. They work by vibrating the brush head at sonic speeds. Though there are more studies done on oscillating toothbrushes, there have been a few done on the effectiveness of sonic toothbrushes. This 2015 study4 found that sonic toothbrushes were significantly better at reducing plaque than manual toothbrushes. An earlier study published in 20145 also found that sonic toothbrushes were more effective at plaque removal. They also found that for a short time (1 week), the sonic toothbrushes were better at reducing the risk of gingival inflammation. After 3 weeks though, the amount of papillary bleeding (bleeding in some parts of the mouth) were the same between those who used sonic toothbrushes and manual toothbrushes. Overall though, it seems that sonic toothbrushes are also better for your oral health than manual toothbrushes.
So we can see that both main types of electric toothbrushes are better than manual toothbrushes. But which is better, sonic toothbrushes or oscillating toothbrushes? In fact, a handful of studies has been done on this topic.
A 2013 study6 tested the two types of powered toothbrushes. Both the sonic and the oscillating toothbrushes were able to significantly reduce plaque and risk of gingivitis, but the results showed the oscillating toothbrush performed better than the sonic toothbrush. Three 2014 studies7, 8, 9 agree with the conclusion that oscillating toothbrushes are better at reducing plaque and gingival bleeding than sonic toothbrushes. All of those studies were performed very similarly and had some of the same researchers between them. From these results, we can see that oscillating toothbrushes are superior to sonic toothbrushes.
For both electric toothbrushes, we can see that risk of gingivitis and the amount of plaque on the teeth are reduced. You did ask more thing which I will answer more directly.
Are there really any long-term benefits associated with electric toothbrushes?
Well, from the trials I've mentioned, we can see that this is most likely the case. The longest trial I directly mentioned2 was 9 months long and it showed that 9 months of consistently using an electric toothbrush did help reduce plaque and gingival bleeding. Manual toothbrush can keep your teeth clean, but it takes more work than an electric toothbrush.
Why doesn't everyone use an electric toothbrush if they are proven to be better?
Well, there are a few things that come into play10. Even though electric toothbrushes are more effective at cleaning teeth and they don't take as much work as manual toothbrush, some people aren't willing to pay so much for a toothbrush. Some of the cheapest electric toothbrushes cost only $5, but the higher quality ones can go for over $100. You can get a five pack of manual toothbrushes for under $5. Another con of electric toothbrushes is that they are a hassle. They are bigger, and they require batteries or they have to be charged. Some people just don't want to deal with that. Lastly, electric toothbrushes can be easily broken. If you drop it, that might be it and you just lost $10. Now, electric toothbrushes are becoming more popular, but some of these cons are still holding people back.
 Powered/electric toothbrushes compared to manual toothbrushes for maintaining oral health
 PComparison of the use of different modes of mechanical oral hygiene in prevention of plaque and gingivitis
 Manual orthodontic vs. oscillating-rotating electric toothbrush in orthodontic patients: a randomised clinical trial
 Is a new sonic toothbrush more effective in plaque removal than a manual toothbrush?
 Toothbrush efficacy for plaque removal
 A 12-week clinical comparison of an oscillating-rotating power brush versus a marketed sonic brush with self-adjusting technology in reducing plaque and gingivitis
 A randomized clinical trial evaluating gingivitis and plaque reduction of an oscillating-rotating power brush with a new brush head with angled bristles versus a marketed sonic brush with self-adjusting technology
 A randomized 12-week clinical comparison of an oscillating-rotating toothbrush to a new sonic brush in the reduction of gingivitis and plaque
 A six-week clinical evaluation of the plaque and gingivitis efficacy of an oscillating-rotating power toothbrush with a novel brush head utilizing angled CrissCross bristles versus a sonic toothbrush
 Manual Toothbrush VS Electric Toothbrush: Pros and Cons