I was browsing this site http://www.fastwheelsets.com/resources-2/ and it's talking about 2 different sets of tires. Is there a necessity to have both tires or I can go fast enough with my training tires ..... do you really feel a difference with the racing tires?
The difference in rolling resistance is small but can be seen in time trial results and maybe in mass start races. The more noticeable difference is that typically race tires have smoother ride because the thinner casing transmits less vibration.
Note that the categories are not clear and one manufacturer's tire marketed as training tire may have less rolling resistance than other's fastest race tire. One web site to check is http://www.bicyclerollingresistance.com/.
Is there a necessity? If you are racing and it is necessary to do everything possible to improve your placement.
Training tyres are somewhere in the product line below race tyres but above economy tyres.
They should cost around the same as a touring or commuting tyre, cost less than a race tyre, and will cost more than a budget tyre.
Training tyres will have more durability than a race tyre, but will be slightly heavier. The idea is to save your valuable race rubber for the race itself, but still ride on rubber that has similar characteristics. That consistency is why tyre makers pair up training models with race models.
Personally I have a puncture-resistant touring tyre on the rear of my road bike, because flats suck on a race. I have a $50 training tyre on the front, because its good enough for the little racing I do.
If you're weight-weening then saving weight on the wheels is fine, but I can't justify the steep price increases for the minimal gains.
I'd like to introduce another axis: Tire material. Soft grippy material vs. Hard less grippy material. The Soft grippy material has the advantages on grippiness, smoother riding, less rolling resistance, but has the disadvantages of getting more flats (for the same thickness of material) and of wearing out faster. The reason for getting more flats for the same thickness of material is that the less grippy tire is more likely to deflect a sharp object, and is harder to penetrate.
Generally speaking, racing tires tend to use soft grippy material, and not much of it, to keep the tires light. Conversely, training tires tend to use the harder less grippy material, and lay it on thicker.
Racing tires therefore grip better, are lighter, and ride smoother. The only problem is they wear out faster. If your are a light body weight short distance rider (going hard for an hour, say) riding on smoother roads that may be acceptable as you wear them out in 6 months. But if you are a heavy body weight rider doing a lot of long distance riding on rough roads, you may be wearing out those tires every month or two, and that gets expensive.
An alternative is a grippy softer tire with thicker material. Maybe 100 grams heavier than a racing tire, but just as grippy and smooth and lasts twice as long.
There are two very popular road tires which together I see on over half of all bikes around where in live in California where the roads are not smooth. One is the thicker soft grippy tire, one is the harder less-grippy tire. Both are made by the same company. (I won't mention the company or the tire names as I think that violates rules). I've used both, and harder tire lasts about twice as long. But I prefer the softer tire for comfort, and I believe it is faster as well. The only (semi-)competitive riding I do is long distance (generally over 100 miles).
Yet another axis is the presence of a puncture resistant band built into the tire. There is a huge variation in effectiveness of these bands, but the really effective bands tend to make the ride very hard and increase rolling resistance. I've used a pair such "puncture-proof" tires for mixed terrain riding - only one puncture where a goats head came in from the side bypassing the band. Apart from that months without puncture, even though the rubber on the tires looked as though it had been attacked by a mountain lion - cuts and deep scratches everywhere.
Nowadays there are also tubeless tires. But that's beyond the scope of my comment.