"Objective facts..."
First, I don't have an NT1-A, so I cannot comment on it. I do have a B1 and a C1, though... and have actually made some measurements of the actual frequency response of those samples that I happen to own. Look for my thread on the Oktava mics from GC here:
http://www.homerec.com/bbs/showthread.php?s=&threadid=101544
Second, on the issue of putting numbers to why mics sound different, there are many factors, as has already been mentioned. None of these fully describe what we hear, but many can be understood scientifically. I'll make an attempt at explaining some of the most important ones in “layman's” terms (as far as I understand them

-- but now re-reading what I put below, I may have failed utterly in doing so :-(
*** The frequency response of a mic shows how it amplifies/attenuates different frequencies. Usually, only omni mics can have fully flat responses. All other mics (and most real-world omni mics) have peaks and valleys, and how the peaks and valleys are in the graph has a big influence on the sound: It can be present, muddy, warm, having air, sizzle, etc. Some of these terms implies certain frequency areas that are being attenuated.
You can achieve a bit similar effect by playing with a good EQ, but all EQ's have side effects as well, so you cannot really make one mic sound just like another by EQ'ing... although in some cases, you can come close.
*** Another factor is the transient response (impulse response). This is how the mic “behaves” when it sees a transient in the sound, such as a click with a lot of high frequency content. If you measure the transient response of a mic (or any other system), it actually means you can derive the frequency response also! Why is this? Well, a “perfect” click (called a delta function) is a signal that contains all frequencies. If you could record such a click, you would get the “transfer function” of the system being measured, here roughly equivalent to the transient response. By doing a FFT of this, you will get the frequency response.
What about the other way round? Well, in order to get to the transient response of a system, you need to know not only how the frequency amplitude response is, but also the frequency phase response, which is how the phase is distorted for different frequencies. And that information is not usually shown by any vendor...
*** A third factor is the distortion of a microphone. This comes in at least two flavors: harmonic distortion, and what is not harmonic. If you send a pure sine wave (which has no overtones) through a mic, you would expect to hear only that. But due to many factors, the sine wave will not look fully like a perfect sine wave after wards. One commonly seen phenomena is that the top of the signal gets slightly compressed. Physically, what happens is that different over-tones (harmonics) is added to the original signal. In the case of a very distorted signal, you may even see a square wave – which is a signal with a lot of overtones added to it.
[Incidently, this is why square, triangle and saw waveforms are popular on subtractive synthezisers. There are a lot of overtones that can be filtered by your nice analog filters, and that gives the “character” of the sound. If you used sine waves as the fundamental waveform here, the usage of filters would be more or less in vain, and you would get a boring sound.]
Back to mics: Some harmonics sound pleasant to us, some don't. The even-ordered harmonics (i.e base frequency multiplied by 2, 4, 6, etc) are usually nice. These are generated by tube equipment. So, when you record through a tube, you add some nice sounding harmonics, due to the harmonic distortion.
Now, no real sound sources generates pure sine waves. There are always overtones. The relationship between the overtones (i.e. how much there is of each multiple of the base frequency) is an important part of helping your ear distinguish between a pan flute and a distorted guitar. Drums are different, since they have less pure frequencies, but I won't elaborate on that now.
What happens then if a real instrument is recorded? Well, for each frequency existing in the original signal (including the original overtones) new harmonics will be added. All will be a multiple of the fundamental (or root) frequency. There is no real difference between the ones from the sound source and the ones you add... so, by e.g. recording through a tube, you change the sound character. This is actually distortion, and in some cases, that is what you look for.
I have tested a B1 and a TB1, for instance. The TB1 sounded a slight bit nicer on my acoustic guitar, since it added some pleasantly sounding harmonics – just a tiny bit, but enough for the ear to detect it. Same applied on my voice. Another extreme of this is a distorted guitar, where the sole purpose is to get a pleasantly sounding distortion from your Marshall stack
So harmonic distortion is often good in very small quantities – and in some cases even in large quantities

But the more you add, the more it will change the character of your sound. If you have a really nice acoustic guitar, you might want to record it as faithfully as possible!
*** One other kind of distortion is adding “noise” of different kinds. It if happens at frequencies that were not there originally (or even at certain harmonics, usually un-even harmonics), it does not sound nice.
*** A third kind of distortion is also important to understand. If you have to frequencies at the same time that are not related, you might get inter-modulation or cross-modulation. TO make a long story short, it means that you suddenly add a frequency that was never there originally, and is not in some “nice” relation to the original frequencies. This is almost always bad! If you have, say, a 2 kHz tone and a 2.3 kHz tone, they may cross-modulate, and you then suddenly start getting a 300 Hz tone as well.
You ear usually does this for you, btw... and is one part of how the ear detects low frequencies. If a deep bass at, say 30 Hz, is present, there will be also 60 Hz, 90 Hz, 120 Hz, 150 Hz, etc present. Now, if even if you remove the 30 Hz completely, the ear will think it heard the 30 Hz signal, if it detects 60 Hz, 90 Hz, 120 Hz, 150 Hz, etc. This can be used to “cheat” the aural system to hear deeper sounds than what the speakers really can produce. And that is why a “straight” bass tone in some cases may be less interesting than one with a lot of overtones (which you may get by recording through a tube amp).
*** Still another factor that is not fully described by the frequency response is resonance. Resonance will certainly show up as a peak in the frequency response, but it will also make its presence known by adding certain frequencies even when none of those were present in the original sound. Think about a drum, for one second. When you hit it, you hear its resonance. But a drum also sounds when you hit other drums nearby it, or if the bass player hits an annoying note that makes your entire kit rattle... Same can be heard with a microphone. The MK-219 is one famous example of an annoying resonance, I've been told.
Hope some of you found this interesting...
-- Per.