Yes and no.

The calories are the same. Once your body has converted the consumable compounds to a common hub comound (say acetate) then there is no difference between the foods that were converted, whether protein, fat, or carbohydrate.

However, not all food compounds are equaly well absorbed, assimilated and converted. Many starches are poorlly digestable, and so are long-chain fatty acids and tanned proteins. Not all foodstuffs that contain those compounds are equally easily digested. A great deal of your unrefined foods passes right through your gut in sheaths of collagen, cellulose, and similar poorly digestable natural plastics. Even animals that can digest that protection with the help of microbes usually need to digest the food twice, either by fermenting it more than once on the first pass through, or by eating it again once it has passed through. Coprophagy reigns OK! Ask any rabbit or termite for example.

 Give a peckish bunny a bash at cat poo, and you will see what I mean. 

Bon appetit or happy dieting, depending on your preferences,

Jon

As a follow-up to Jon's comment on varying efficiency of digestion. The calorie values given on most food usually take this into account and are lower than the calorie value one would obtain by burning the food and measuring the heat output. 

Perhaps worth considering the effect of sugars and carbohydrates on blood sugar levels, ie in foods that release energy very fast, often referred to as foods with high glycemic index value like refined carbohydrates.

If blood sugar rises too fast and high after consumption then the bodies blood sugar balancing act kicks in to remove sugars from the blood and store them as fat.

When the rate of absorption through digestion is high, eg with calorie sources that have a simple mechanism then the chance of raising sugar levels too high increases. Varied calorie food types that take varying times to breakdown to useful energy will smooth the release of energy into the body over a period and reduce the likelihood of the need for rapid sugar-fat conversion.

1000 cals of sugar could also make it easier to increase consumption. Think of the post Chinese meal munchies that happen when the sugar level drops as the insulin removes too much of it and triggers initiates a hunger response. 

The complex role of appetite based on the interaction with blood sugar, fat and satisfaction/hunger signals has until recently been largely ignored for the simpler model of pure calorie counting. This makes for interesting reading:

http://www.time.com/time/specials/2007/article/0,28804,1626795_1627112_1626457,00.html

Perhaps worth considering the effect of sugars and carbohydrates on blood sugar levels, ie in foods that release energy very fast, often referred to as foods with high glycemic index value like refined carbohydrates.

If blood sugar rises too fast and high after consumption then the bodies blood sugar balancing act kicks in to remove sugars from the blood and store them as fat.

When the rate of absorption through digestion is high, eg with calorie sources that have a simple mechanism then the chance of raising sugar levels too high increases. Varied calorie food types that take varying times to breakdown to useful energy will smooth the release of energy into the body over a period and reduce the likelihood of the need for rapid sugar-fat conversion.

1000 cals of sugar could also make it easier to increase consumption. Think of the post Chinese meal munchies that happen when the sugar level drops as the insulin removes too much of it and triggers initiates a hunger response. 

The complex role of appetite based on the interaction with blood sugar, fat and satisfaction/hunger signals has until recently been largely ignored for the simpler model of pure calorie counting. This makes for interesting reading:

http://www.time.com/time/specials/2007/article/0,28804,1626795_1627112_1626457,00.html

No.  If you restrict the number of calories in your diet, you will not show any difference in weight loss whether or not your restrict calories from fat or calories from carbohydrates.  This has been studied fairly extensively, and there's no evidence there's any difference in actual weight loss. 

This figure is from a paper entitled "Comparison of Weight-Loss Diets with Different Compositions of Fat, Protein, and Carbohydrates."  Each different color represents a diet.  The difference between two diets are significant (i.e. not due to random effects) if the bars around the colored dot do not overlap.  You can see that all of the bars overlap, meaning that there was no significant difference in weight loss or in waist circumfrence between any of the diets.

However, this does not mean you will experience the same success in a diet which restricts fat as you would in one that restricts carbohydrates.  Our bodies react in different ways to eating different foods, and fats help us feel satiated.  In these studies, participants ate and only ate what the researchers gave them to eat, but you will not have this type of enforcement.

Research suggests that a balanced calorie restricted diet is the best way to diet.  Too little fat and you will often feel hungry; too little carbohydrates and you will feel tired and weak.  Also, keep in mind that high fat/high protein diets place you at a greater risk for heart disease.  The most important part of losing weight if you are obese or overweight is to better your health, and such a diet runs you the risk of losing those benefits.