What is nitrogen?
Nitrogen is an ‘inert’ gas which means it is a property-free, non-reactive substance or more simply put ‘nothing’. With nitrogen, actually nothing is exactly filled - nothing but pressure.

Advantages of nitrogen
The good old regular air that we have been using to inflate automotive tyres for the past 100 years is actually 79.1% nitrogen, 20% oxygen and 1% other gases. The problem with regular air is that it also contains about 20% oxygen. Although oxygen is essential for survival, the process of ‘oxidation’ causes rust and corrosion and is harmful to rubber and steel in tyres and wheels.

Since nitrogen is an inert gas, oxidation can be avoided with its use. 
Further, without the presence of oxygen or ‘O’ in ‘H2O’ - water or condensation cannot form in a tyre. Hence, without any oxygen there is no oxidation or moisture for the tyre to rot or the wheel to rust. 
Moreover, oxygen has smaller molecular structure. A molecule of oxygen is roughly 1/4 of the size of a molecule of nitrogen. Oxygen molecules are so small that it is completely normal for air filled tyres to lose air by ‘permeation’. All tyres have microscopic pores through which any inflating gas, air and nitrogen included, will seep out by the process of permeation over an extended period of time, gradually lowering the inflation pressure. Nitrogen has larger molecules than air and will permeate through the tyre more slowly than air, thus maintaining the inflation pressure longer. It is generally assured by the nitrogen fillers that the tyres would retain constant pressure for 3 months. However, in reality, we found that the nitrogen remained intact for almost 2 months. 
The frequent drop in tyres’ air pressure increases the rolling resistance, which in turn causes the decrease in mileage. Hence, if you do not have the habit of checking the tyre pressure regularly, nitrogen would be recommended, as the tyres will retain proper operating pressure much longer than air.
For the advantages nitrogen offers, all vehicles subject to extreme temperature conditions such as race cars and aeroplanes are filled with nitrogen. The use of nitrogen allows better control over the tyre pressure in extreme conditions. The tyres also run cooler, which is especially important in racing because the grip of race tyres is highly dependent on tyre temperature.
Thus on long drives on highways, where runs are more with higher speeds, nitrogen is beneficial because the tyre pressures do not fluctuate resulting in better control and safety.

Mixing air and nitrogen in tyres
There is no problem if you mix air and nitrogen. If the tyre has a puncture or a situation when there is a drop in one or more tyres, you can fill them with regular air, since you won’t get nitrogen everywhere. You may also top up nitrogen filled tyres with normal air. There is no ill effect to the tyre or your vehicle’s handling. Later, when convenient or if you want to change over to nitrogen at all, purge the air mixed tyre and refill with nitrogen once again. It is also not a problem if one out of the four tyres is filled with air. However, the problem would be that the air filled tyre may have pressure fluctuations in the long run and behave differently.

Disadvantages of nitrogen
Cost and convenience
The main problem with nitrogen is that it is not commonly available in Indian cities whereas air for tyre inflation is universally available. Moreover, the latter is available at a very low cost. Just about any service station or tyre dealer will fill up your tyres or let you adjust your tyre pressure for free or against a trifle amount. But that is not the case with nitrogen which generally costs Rs 100-120 for 4 sedan tyres in Kolkata.

Worn out tyres
If your tyres are old or are not in a good condition, avoid filling them up with nitrogen. This is because the nitrogen would then leak out through faulty valves or the seal between the tyre and wheel rim making the entire exercise futile.

Nitrogen quality
It has also been found that the quality of nitrogen varies between outlets resulting in no derived benefit. Thus, if you find that the tyre pressure with nitrogen is fluctuating within a few days of refill, either change your nitrogen filler or check for leakage in your tyre before the next refill. 
In conclusion, consider the use, convenience and cost before you decide on either air or nitrogen.