Low carb (LC) diets are often based on the role of insulin in fat synthesis (de novo lipogenesis), esterification (fatty acids >> triglycerides) and storage, while inhibiting β-oxidation. The most basic argument is that carbs increase insulin and insulin increases fat storage, which is the carbohydrate-insulin-hypothesis (CIH) of obesity.
Low fat (LF) diets don’t rely on any scientific mechanism besides the FLTD and the main argument is that fat has more calories per gram than carbs or protein, therefore high fat foods have more calories (not necessarily) and reduced fat foods have fewer calories. Just like CICO, LF completely neglects the hormonal regulation of appetite.
The CIH suggests a LF,HC diet would be the most obesogenic, while the LF,CICO suggests a LC,HF diet would be the most obesogenic. However, people on both LC and LF diets spontaneously reduce their calories and lose more weight than those on calorie restricted diets . This suggests neither macronutrient is obesogenic per se and that there are other factors involved.
One likely factor is that LC and LF diets limit junk foods. Junk foods are generally low in protein and nutrients, but high in fat, carbs and toxins such as artificial trans fats. Junk foods are too high in carbs for LC and too high in fat for LF and so are avoided on those diets. When LC or LF dieters replace junk food they are probably improving food quality by increasing protein and nutrient density and reducing artificial trans fats. Protein is very satiating and low protein diets (<15%) can increase total calorie intake , multivitamin and multimineral supplements have a modest effect on weight loss  and artificial trans fats can increase weight gain in an isocaloric diet .
Evidence against the CIH of obesity
- Insulin isn’t the only hormone involved in fat storage. Acylation stimulating hormone has the same functions as insulin on adipocytes and is released in response to dietary fat 
- Insulin doesn’t lock fat away and increase appetite (the insulin >> cellular starvation argument). Insulin is a satiety hormone and satiety from protein correlates with the insulin released 
- High fasting insulin doesn’t significantly block fat oxidation and reduce metabolic rate in obesity. People with obesity have equal or more free fatty acids than lean people because the obese have insulin resistant adipocytes and high fasting insulin is associated with a higher metabolic rate, independent of weight 
- Hyperinsulinemia isn’t a causal factor in obesity. Diet induced obesity in mice without certain genes for inflammation increases weight gain, but not insulin resistance and hyperinsulinemia . Liver-specific insulin receptor knock out (LIRKO) mice are insulin resistant and have hyperinsulinemia, but actually weigh slightly less than normal mice 
The CIH seems to have merged its message with low fructose. The idea now being that fructose is metabolised like ethanol. Fructose increases liver insulin resistance, which in the context of a high carb diet results in hyperinsulinemia and high insulin blocks leptin, increasing leptin resistance. The new CIH/LFruc calls fructose a toxin.
Fructose is metabolised like alcohol: Fructose and ethanol metabolism are very different and ethanol isn’t a carbohydrate. Fructolysis is actually very similar to glycolysis as both involve the splitting of fructose/glucose into 2 pyruvate (pyruvate can enter the mitochondria and help form acetyl-CoA) and fructose/glucose can be converted to glycogen during fructolysis/glycolysis. The important difference is that phosphofructokinase regulates glycolysis, but fructolysis doesn’t have an equivalent regulator . Alcohol dehydrogenase converts ethanol to acetaldehyde (which produces oxidative stress and is highly toxic ), and acetaldehyde dehydrogenase converts acetaldehyde to acetic acid (vinegar), which gets converted to acetate and finally helps form acetyl-CoA. Fructose metabolism is very different to alcohol, fructose can be converted to liver glycogen and does not produce highly toxic metabolites such as acetaldehyde.
High insulin blocks leptin, causing leptin resistance: LIRKO mice refute this argument as they have extremely high circulating insulin levels, but weigh slightly less than normal mice . Insulin actually increases leptin transport across the blood brain barrier .
Fructose is a toxin: Ethanol and acetaldehyde are toxins, but fructose isn’t necessarily. The relationship between fructose and toxicity is likely a threshold effect. Fructose probably only becomes a problem when the liver mitochondria become overwhelmed by excess acetyl-CoA (energy overload). Until then fructose can refill liver glycogen and be metabolised in a similar way as glucose. The same thing is true for other sources of calories – they aren’t generally a problem unless someone is overfeeding. The difference between fructose and other calories is that the threshold effect is probably more sensitive with fructose because it has no negative regulator of its metabolism (such as phosphofructokinase) and a lot of it (~50%) is metabolised by the liver* .
Much of the discussion on the harmful effects of fructose has been extrapolated from studies using superphysiological doses of fructose (~60% of total calories). At normal doses in humans fructose doesn’t substantially increase de novo lipogenesis**, deplete ATP to uric acid, increase blood pressure or weight gain .
Low carb diets work, but it’s not because of insulin. The high refined sugar SAD is obesogenic, but not because fructose is a toxin, fructose metabolism is like ethanol, or because hyperinsulinemia increases leptin resistance.
* When we consider that fructose toxicity depends on conditions and context, we can better understand how some people do fine on a fruitarian diet. Even with fibre, which slows the rate of fructose entering the liver, fruitarian livers deal with high fructose load, probably much higher than those on the SAD. Calorie restriction and/or endurance exercise seem to be common among fruitarians. Calorie restriction reduces the load and both increase the rate of aerobic metabolism   and mitochondrial biogenesis  . But also, there is more the fruit than fructose. Fruit contains many nutrients and other beneficial compounds.
** If one of the reasons fructose is bad and fattening is because de novo lipogenesis can convert it to fat, then dietary fat should be even worse
(1) The Carbohydrate Hypothesis of Obesity: a Critical Examination
(2) Is Insulin Resistance Really Making Us Fat?
(3) Fate of fructose: Interview with Dr. John Sievenpiper