Vedic Mathematics is often dubbed as the 'Yoga' of mathematical thoughts. This is because, its unique techniques encourage mental calculations, and provides mechanisms that aid in enhancing interest and understanding of numbers and mathematical concepts.
The most common apprehension in using the 'unconventional tricks' of Vedic Mathematics, is that the solutions look too magical to believe that they would actually work (in all cases). More importantly, they seem to be devoid of 'mathematical logic'  whatever that means!
As a simple illustration to introduce Vedic Mathematics, let us consider squaring 99:
Since, 99 is close to (slightly lower than) a power of 10 (100, in this case), let us use Upasūtra:
यावदूनं तावदूनीकृत्य वर्गं च योजयेत्
yāvadūnam tāvadūnīkṛtya vargañca yojayet
(Lessen by the deficiency, and set up square of the deficiency)  for 99
^{2}, in simple steps:

Steps 
99^{2} 
1. 
Consider the nearest power of 10 as the Base 
In this case, the Base, closest to 99, is 100. 
2. 
Find the deficiency 
In this case, the Deficiency = 100  99 = 1 
3. 
Subtract the deficiency from that number 
In this case, 99  1 = 98 
4. 
Setup as many zeroes, as the digits of the number 
In this case, 99 being a 2digit number, we get 9800 
5. 
Square the Deficiency, and add to get the answer 
1 being the Deficiency, and 1^{2} = 1 9800 + 1 = 9801, which is the answer! 
So, for a practitioner of Vedic Mathematics, for something like 996
^{2}:

Clearly, the deficiency is 4, and 4^{2} = 16
Also, 996  4 = 992
And, 992,000 + 16 = 992,016
Thus, 996^{2} = 992,016 
Of course, it is so amazingly simple that it can be done mentally, with some practice  within a few seconds. Taking it ahead, for something like 99988
^{2}:

Clearly, the deficiency is 12, and 12^{2} = 144
Also, 99988  12 = 99976
And, 99976,00000 + 144 = 99976,00144
Thus, 99988^{2} = 9,997,600,144 
Now, that should be magical for somebody who is newly introduced to Vedic Mathematics.
The irony is: Most of us, knowingly or unknowingly, have been learning and using the 'tricks' of Vedic Mathematics through preparations of various competitive examinations  albeit, at a later stage, and during a race against time.
Speed does matter. Especially in today's life, when the results can be obtained faster than typing the digits in a calculator.
But, not just speed  the real power of Vedic Mathematics is in calculating the seemingly impossible calculation  that too, with ease. Let us consider the above example for a 20digit number, say 99999999999999999998
^{2}:

All we have to do is stay calm, and have faith in Vedic Mathematics.
Clearly, the Deficiency is 2, and 2^{2} = 4
Also, 99999999999999999998  2 = 99999999999999999996
(The operation is actually on the last digit)
And, 99999999999999999996, 00000000000000000000 + 4
= 99999999999999999996, 00000000000000000004
(The operation is again on the last digit, of 20 trailing zeroes)
Thus, 99999999999999999998^{2} =
9,999,999,999,999,999,999, 600,000,000,000,000,000,004 
Don't bother to check with a calculator, we don't expect it to work.
This is the reason that Vedic Maths, more often than not, is promoted as an alternate mathematical methodology which is 'superior' to the conventional methods of calculations, or tricks, and even mental maths for 'fastest' mathematical calculations that brings out the 'human calculator' within oneself!
It is a fact that steps of conventional methods are too tedious to be performed mentally, which is why Vedic Mathematics is the source of thousands of 'Mental Maths' books/tutorials/videos doing the rounds  not to mention the innumerable 'tricks', 'shortcuts' and 'secrets' that educational 'gurus' demonstrate to provide their students an edge in competitive examinations, all over the world.
And considering the fact, that the above example is based (only) on an Upasūtra  not even a Sūtra  should provide a fair idea of what Vedic Mathematics is about.
But, why does it work? For the above Upasūtra, let us consider the following:

Assuming N is a number close (and less) to a power of 10, then N = a  b
'a' being the power of 10, and 'b' being the Deficiency
As in examples above, 99=1001, 996=10004, and 99988=10000012
Now, N^{2} = (a  b)^{2}
= a^{2}  2ab + b^{2}
= a (a  2b) + b^{2}
But, N = a  b. So, substituting a = N + b
N^{2} = a ([N + b]  2b) + b^{2}
= a (N  b) + b^{2}
This is exactly what this Upasūtra makes us do:
99^{2} = [100 × (99  1)] + 1^{2} = 9800 + 1 = 9801
996^{2} = [1000 × (996  4)] + 4^{2} = 992000 + 16 = 992,016
99988^{2} = [100000 × (99988  12)] + 12^{2} = 99976,00000 + 144 = 997,600,144

Needless to state that the principles of Vedic Mathematics work because they are mathematical.
The above Upasūtra, being mathematical, would work even for numbers not close to the Base, but that will not be very smart  especially when there are other methods (like Upasūtra:
dvaṅdvayoga 
discussed here ») to do the same. For example, using this Upasūtra for 77
^{2} would require us to find 23
^{2} (23 being the Deficiency). That would be foolishness, and not failure of Vedic Mathematics.
Obviously, further explanations and details (including the number being close to, but greater than the Base) are presented in the respective Upasūtra:
yāvadūnam tāvadūnīkṛtya vargañca yojayet (
discussed here »), but the point is: Vedic Maths provide no magical trick, and is certainly not devoid of 'mathematical logic'. Fact is, it is practical step ahead of the 'Algebraic Rules'  that we are comfortable with.
The standpoint typical to Vedic Scholars is that the
Vedic Civilization was aware of the Algebraic Rules, and generalized them for practical usage with easytoremember phrases (read Sūtras). But, we don't want to get into that, and just stick to the mathematical structure.
Having stated the above, irrespective of everything, Vedic Maths can indeed lead to amazingly fast calculations, with some practice, for the mathematically inclined. There is no denying that the principles of Vedic Maths provide methods to convert cumbersome and complicated calculations into simple steps, which can be mentally executed to reach the solution at ease. However,
 Vedic Mathematics is not an alternate methodology:
Even though, it seems to provide alternate methods of computation  Vedic Mathematics is a way of moving ahead with the Algebraic Rules. Vedic Mathematics is (more of a) practical nextstep of the Algebraic Rules, and should be viewed as a layer above it.
 The principles of Vedic Mathematics are not tricks:
Tricks work on specific situations and cases. When a 'rule' is general, and may be equally applied on all cases, without exception  it is called a 'formula' or 'theorem'. Labeling Vedic Mathematics as a set of tricks would be demeaning to a set of excellent formulae  irrespective of its origin, influence, or even religious importance.
 Vedic Mathematics is not Mental Maths:
Although the principles of Vedic Mathematics are easy to use, the usage of the principles mentally requires practice and a certain amount of mathematical skill  as in any other form of mathematics. Vedic Mathematical principles being easier, make it comparatively easy to execute them mentally. This does not mean that everybody can become 'human calculators' by learning them.
 Vedic Mathematics should not replace the conventional methods:
Replacing conventional methods with the principles of Vedic Mathematics at school, or otherwise, would be like learning the Vedas instead of science. Even teaching both methods sidebyside to children would amount to confusion and unnecessarily burden them. The best age to learn Vedic Mathematics is from age 1415 years onwards, preferably for the mathematically inclined.
 The principles of Vedic Mathematics are not 'more correct' or 'superior':
In science and mathematics, there may be more than one method to reach a solution. This does not mean that one of the methods is 'more correct' or 'superior' than the other just because it is fast, easy, can be mentally executed, or even because the results can be derived from left to right!
 Vedic Mathematics do not cover the entire sphere of mathematics:
Mathematics is a far bigger subject than what is covered by the principles of Vedic Mathematics. Vedic Maths find its best applications with real numbers, in real life scenarios (when one doesn't have penpaper or a calculator), competitive examinations (when the derivation is not as important as the answer), practical problems (when one needs a calculated value as an input for some other task), and verifying calculations (when one can use Vedic Maths to quickly check a calculated figure already calculated using conventional methods).
Most of our above standpoints are opposed to hardcore 'Vedic Mathematics' (and even Hindutva) fanatics. Nevertheless, this web endeavour should speak volumes of our love, appreciation and respect for Vedic Mathematics and its amazing principles.
All said and done, there is no denying that Vedic Mathematics is far more than 'wise shortcuts'. It provides techniques that encourage us to execute the computations mentally  thereby stretching and excercising our capability to deal with mathematical thoughts.
The techniques improve our overall mathematical capability, by clarifying the positional notation (as in
śūddha,
discussed here »), by making us visualize the numbers to mentally execute calculations (as in
ūrdhva tiryagbhyāṃ,
discussed here »), by helping us to think closely with the number system (as in
ānurūpyeṇa,
discussed here »), and by forcing us to anticipate consecutive steps (as in
dhvajāṅka,
discussed here »)  to list a few positive influences.
Vedic Mathematics is truly the 'Yoga' of mathematical thoughts, and its use can help overcome 'Maths Phobia' by making one 'play' with numbers and mathematical concepts.