I would note that there are actually two very different types of tantalum capacitors, and information about one is not applicable to the other.
NASA did extensive research on wet tantalum capacitors which are not the same thing as the widely available SMD tantalum capacitors and anything said about wet tantalum capacitors is irrelevant to the more familiar ones you are probably talking about.
The primary usage case for the commercial plastic encapsulated SMD or plastic dip leaded tantalum capacitors we're probably all familiar with really comes down to two things:
When you need a lot of capacitance in the smallest volume possible. Tantalum capacitors 3 times the theoretical volumetric efficiency of the next best capacitor type for this, aluminum electrolytic. In practice, the margin is actually even more. Case in point: the sintered nugget of tantalum inside a 220uF 6.3V capacitor is only 1.6 cubic millimeters (0.0016mL). The plastic packaging itself takes up most of the space, not the actual capacitive element.
When you need longevity. Tantalum capacitors don't suffer dielectric degradation when stored discharged for too long like aluminum electrolytics will, nor do they dry out like aluminum electrolytic capacitors do (and they do sometimes very quickly if they get hot, most standard aluminum electrolytic capacitors are only rated for 2000h of service life at 85C).
Tantalum capacitors, simply put, have the highest volumetric efficiency of any capacitor type currently available, and when used correctly, will continue to work for a very long time even in high ambient temperatures.
As a designer, those are really the only two reasons one might choose to use them in modern designs. The polymer versions of them, while expensive, have excellent combinations of high capacitance, and ripple current capability in a small space, while tolerating constant heat much better than any other type of polymer capacitor. However, the often relatively high ESR of the cheaper standard types are actually often an advantage as it will limit in-rush current (which, with other capacitor types such as ceramic, even just 10uF can cause voltage spikes of 20-40V due to the brief inrush current and a power cable's self inductance), or helps to stabilize certain LDO Voltage regulators.
For many, they are often the quintessential lazy design choice. They're small, I know they will be good enough where they need to be good, and bad enough where they need to be bad, to be fairly safe to put in one or two spots in a design and be confident that they won't cause any unexpected problems ( unlike ceramic capacitors for example).
That said, they do have some disadvantages. And because of careless design, actually have a somewhat deserved reputation for being unreliable and having a propensity for fire.
Have you heard of thermite? It is a powder made of (at least) iron oxide and aluminum. It is flammable, but has a very high ignition temperature beyond what most flames can achieve, making it relatively safe and an important industrial tool. When ignited however, it burns incredibly hot - hitting 2500 °C or more. It is an reduction-oxidation reaction, with the oxygen in the iron oxide forming aluminum oxide and precipitating pure elemental (and definitely molten) iron.
Tantalum capacitors are typically constructed using tantalum metal and manganese dioxide, which will happily undergo this same kind of reduction-oxidation reaction as thermite. They need only reach the ignition temperature (which is much lower, around 900 °C) to release a ton of thermal energy evidenced by a jet of flame that briefly shoots out of the capacitor.
This kills the capacitor.
So this can happen in a vacuum, under water, really anywhere you didn't want fire, tantalum capacitors can, if used improperly, ensure you have fire there anyway!
The issue is that they are very sensitive to reverse voltage bias and can tolerate very little and not for very long before the layer of tantalum pentoxide serving as the dielectric barrier will fail, resulting in a short. Now factor in how the ESR is probably an ohm or a few, and you now have that value resistor connected directly across the power rails of your device.
Now recall that the pellet is probably quite small, like 1.6mm cubed for a 220uF capacitor as mentioned earlier. It will have no issues hitting 900 °C.
For this reason, you must be very careful to ensure that tantalum capacitors won't ever see any reverse bias. This can be done through careful design and often at even more cost by adding other components designed to protect the capacitor from reverse bias. If this is done correctly, then tantalum capacitors can last a very long time.
Wet tantalum capacitors are very different beasts and have a liquid electrolyte made of sulfuric acid inside. These are very expensive, prohibitively so, limiting their use to very specific applications, usually aerospace. They can be made to withstand higher voltages than commercial solid tantalum capacitors, and have even higher volumetric efficiency than solid tantalum's already impressive specs in the same.
They also have very different failure modes, they can dry out (but are usually hermetically sealed to prevent this - but it can still a problem). There is a NASA document that very thoroughly investigates them, but none of that information can be applied to the more familiar solid tantalum capacitors.