The main disadvantage to either toe clips or (even moreso) clipless pedals is that until you get used to them you're likely to have a few rather inelegant falls (like pulling up to a stoplight and just falling over).
There are other cases where they can contribute to falls (or exacerbate falls that would have occurred anyway), particularly in off-road conditions (which is one reason that off-road bikers often don't use them).
To me (being partly disabled) the main advantage is that I expend less energy keeping my feet on the pedals.
This isn't as trivial as it sounds:
Without some sort of retention system you must constantly work to keep your feet centered over the pedals -- get off-center and your foot slides off.
Of course, people mostly manage this unconsciously, but it still takes considerable effort/energy, especially when you're tired or struggling against a hill or headwind or whatever.
The ability to "pull" on the pedal and gain energy on the upstroke is probably not significant in the general case, but can be important when you need a brief burst of power/torque.
More important, and somewhat related to the previous paragraph, is the fact that you can extend your power stroke slightly, beginning a few degrees earlier and ending a few degrees later.
You don't get more peak torque, but you do get more energy per unit time.
bicycles.stackexchange.com/users/1584/Daniel R Hicks
There is some disagreement within the cycling community about how beneficial clipless pedals and toe clips are over regular platform pedals.
I won't get into the argument here, but feel free to read the above link for the best thought-out argument against clipless pedals.
However, the most commonly named advantage of clipless pedals and toe clips is the ability to pull up on the back stroke, theoretically creating 360 degrees of power.
As I understand it, this idea hasn't really held up under lab tests.
The additional advantages of toe clips are that they're cheap, easy to get in and out of, offer a large degree of float (the way that your foot moves around on the pedal), and can be worn with almost any shoe.
The disadvantages are that they can scuff up and/or wear holes in your shoes, they're not quite as easy to get out of as a platform pedal, and don't offer as much power through the whole 360 degrees of motion as clipless pedals do.
The additional advantages of clipless pedals are a more secure attachment to the pedal, and the ability to wear a shoe with a completely rigid bottom.
The disadvantages are that the reduced float can contribute to repetitive stress injuries, they require special (and often expensive shoes), and they're more difficult to get out of.
You even see experienced cyclists fall over at stoplights once in a while because the pedal refuses to disengage.
My personal preferences are platforms for downhill, trials, and urban riding, toe clips for commuting, clipless for mountain biking and road racing/training.
bicycles.stackexchange.com/users/4239/jimchristie
They are more efficient.
Firstly you need to do less work to keep your feet on the pedals.
Though it seems like that might be trivial
you do expend energy and or concentration on keeping you feet on the pedals
this will reduce that somewhat.
Additionally in some circumstances it is safer, your feet will not slip off the pedals when you go over a bump.
Finally you will be able to pull on the up stroke using different muscles and giving you more even power.
I had a problem with toe straps that I have not had with proper clip in pedals and shoes.
In order to remove your feet from toe straps, especially ones done too tightly you must take them straight back off the pedals.
This is most unnatural as the reflex action is to move them sideways especially if you are slowing down and staring to lean to one side.
bicycles.stackexchange.com/users/5247/robthewolf
It's surprising that this is as hot a topic as it is.
Clipless pedals are more efficient, and you can get stiffer shoes for them.
Competitor or not, if you step foot into clipless pedals I think you will notice the difference.
Go take a pair for a test ride and you will probably favor the increased control, especially if you ride longer distances.
Perhaps the main downside that can't be overcome one way or another, is the increased cost.
As for that infamous argument about whether or not you can produce more power, hopefully we will have enough real world data to bring the argument to rest.
bicycles.stackexchange.com/users/5237/Scott Hillson
To ride a bike efficiently the ball of your foot must be over the pedal spindle and to answer your question, toe clips and straps are one way of doing this.
Up to the mid-80s that was the method used by all serious cyclists.
You could buy small, medium and large toe clips for any size of shoe to fix your feet in the correct position.
Metal or leather shoe plates with a groove that fitted over the rear pedal plate were nailed to the soles of the leather shoes.
The strap had a quick release buckel and went through the pedal and through a loop on the toe clip.
When the foot engaged the pedal and the strap was pulled tight your foot was held very tight.
To get your foot out
you had to bend down and flick the strap buckle, lift your foot and pull it back.
The modern method is clipless pedals, they serve the same purpose, fixing the ball of your foot always over the pedal spindle.
The initial cost is quite expensive as you require special pedals, shoes and cleats.
When the cleats are fitted to the shoes you just have to push your foot down on the pedal till it clicks and to release it you turn your foot to the left or right.
The holding tension can be adjusted on the pedals to allow for easier entry and release.
They are very good
and I recommend you buy them.
To see and feel the benefit of fixing your feet just try going up a steep hill or sprinting as hard as you can, or ask yourself why do all the Pros use them.
bicycles.stackexchange.com/users/1266/Kipper
I'm considering getting some toe grips or straps for my bike.  What are the benefits of these over normal pedals?
Benefits of toe grips/toe straps
bicycles.stackexchange.com
<pedals><clipless><toe-straps-clips>
bicycles.stackexchange.com/questions/13039
bicycles.stackexchange.com/users/5189/evoandy
There are some advantages to toe grips as they keep your feet firmly on the pedal and allow you to apply more pressure, thus creating more power. They also mean you have to expend less effort keeping your feet on the pedal. There are, nevertheless, some disadvantages, namely they take a little getting used to as you have to pull your foot directly backwards to get it off the pedal. This may result in a few falls at first when you have to make a sudden stop. These disadvantages mean they are not much used ny the MTB community. The main advantages of clipless pedals and toe clips include solid positioning of the feet in an efficient cycling position and possible (but unverified) added power from pulling up on the pedals. Professionals all use clipless pedals, which is a strong vote of confidence in their efficacy. Toe clips and clipless pedals have the potential disadvantage of the need to remove your foot from the fastening, which can cause unnecessary falls or exacerbate accidents. Toe clips have the advantage of being compatible with normal shoes, while clipless pedals allow more rigid soles with a more secure attachment to the sole.
The main advantages of clipless pedals and toe clips include solid positioning of the feet in an efficient cycling position and possible (but unverified) added power from pulling up on the pedals. Professionals all use clipless pedals, which is a strong vote of confidence in their efficacy. Toe clips and clipless pedals have the potential disadvantage of the need to remove your foot from the fastening, which can cause unnecessary falls or exacerbate accidents. Toe clips have the advantage of being compatible with normal shoes, while clipless pedals allow more rigid soles with a more secure attachment to the sole.