Why are porcelain teeth strong? Generally, replacement porcelain teeth (like those used for dental implants) are very strong; double the hardness of enamel (7 as opposed to 5 on the Mohs scale):

However we all know how easily a porcelain vase breaks:

Imagine if the replacement teeth were that weak!
Why is it that the porcelain teeth don't break like their vase counterparts?
 A: There are several things you need to consider.
First - the different between toughness and hardness. When a material is hard, it can sustain a very large stress before elastic deformation takes place. This is what is measured with the Vickers hardness test - which is one of the ways that Mohr's hardness is determined. All porcelain is hard, although different grades of hardness do exist.
The second property is toughness. This relates to fracture initiation and propagation. Fracture typically starts at a point of stress concentration - where the shape of an object makes the local stress greater than the average stress. This is clearly demonstrated with plastic bags containing potato chips, for example: there is often a small "nick" in the plastic, and if you tear the plastic at that point it's easy to rip. This is because the tip of the nick is a point of stress concentration. Try to rip the bag open elsewhere, and it's much harder.
Tough materials typically have two properties: there are not many points where stress concentration can occur, and more importantly, when a crack starts to grow there is something about the structure of the material that stops it from propagating. For example, fibers in a composite material not only add rigidity (stiffness) but the also prevent the propagation of cracks.
Finally, in the example of dropping a porcelain vase, the stress of the impact is greatly amplified by the shape of the vase (impact is some distance from the center of the vase, resulting in a large shear wave; but this wave has to propagate through a thin shell of the vase).
In general, making a ceramic tougher is done by using very fine powders, and using techniques that minimize the presence of bubbles or other stress concentrators. For porcelain teeth, the volume is quite small so one can afford to work carefully to minimize the presence of imperfections. This is harder (and more expensive) to do on a large scale.
An interesting detail pointed out by @starrise in a comment - I'm adding to the answer :

To add to this answer: zirconia (ZrO2) is metastable in the tetragonal phase at STP, and will stay in that phase for very long times as part of a polycrystalline ceramic composite (5% degradation over 10 yr in mouths). Cracks near a t-ZrO2 grain will induce a volume-increasing phase change to a monoclinic structure, inducing compressive stress around the crack tip, increasing energy required to advance the crack. The overall process toughens a polycrystalline ceramic. This method is used to toughen dental porcelain.
  Just in case, and to clarify, it's only metastable "baked in" as part of a polycrystalline structure, whose mechanical constraints keep it in tetragonal form, or when in solid solution with a stabilizing element such as yttrium or cerium. And you're quite welcome! Look up tetragonal zirconia polycrystal (TZP) and/or zirconia toughened alumina (ZTA) for more information

All these things matter. It's not a complete description but I hope it helps a little bit.
A few links:
interesting description of the structure of the tooh and material properties
Letter to Nature on using tetragonal zirconia to achieve tougher ceramics
A: This question reminds me of a joke: "A man riding his motorbike is pulled over by a policeman. - Why are you not wearing helmet? You're supposed to wear one, instead of this knit cap on your head. - The man: Because I think the knit cap is stronger, so it'll protect me better - How so??? - Because I tested them, I went to the 3rd floor and drop both of them to the ground, the helmet was totally broken while this knit cap was still intact!"
Let's not talk about the chemical elements in the vase and in the tooth first. Your question and concept about "toughness", or in general, durability of materials, is so bizarre :) You know the eggshell? Well, it can be strong when the forces are applied from outside (Check this). But the same eggshell should be delicate enough for a chicken to hatch from inside.
You can easily bend a thin plate of steel, but that same steel, in the form of T-fold or wavy fold is not easy to bend.
Now come back to the porcelain, take a broken piece from that vase, and try to apply force on it, press it, squeeze it, can you deform that small piece? Like make it thinner, longer, etc. That will give you some idea of the toughness of porcelain (although porcelain teeth are normally more durable than the porcelain of the vase).
A: White pseudo crowns are the options for the hardened steel crowns.T he progression in the fired innovation has given another alternative for the patients enduring with dental rot. This sort of crowns is conceived from the restrictions of tempered steel crowns, for example, metal sensitivity, indicated when they grin or talk and looks unnaturally. The material utilized in the creation of these crowns is zirconium. It is the most grounded artistic material accessible for the dental practitioners today. 
The fundamental distinction of white pseudo crowns is that it reestablishes the tooth to its typical shape and shading and we can't discover contrast between these teeth and regular teeth. They are stylish in nature. It is the fresher material utilized by the dental specialists. It is hard material and can't fit a wide range of tooth. The greater part of the characteristic tooth ought to be expelled to fit WHITE PEDO CROWNS. The idea of the material is probably going to break. It is likewise more exorbitant when contrasted with the conventional ones. 
In the current circumstance, individuals are searching for the tasteful interest which is the study of magnificence. Distinctive crowns are made to address the issues of the clients. The solidness of the material is dependable and it is biocompatible than some other material utilized in this kind of dental prescriptions. These kind of crowns are minimal costly than the crowns made of tempered steel. These are pre-manufactured fittings. 
The fundamental impediment of these crowns is the point at which they contact with spit or blood they will get slacken at the season of fitting. The primary exemptions in which these sorts of crowns ought not to be utilized is the point at which the tooth is completely rotted and the individuals who have swarmed tooth. In the good 'old days just tempered steel was utilized, however, these days there are numerous alternatives before the patients who ought to be cautious in picking the correct ones which fits the best.The budgetary position ought to likewise be considered in choosing the crowns
