I have noted that the Shia Muslims use a stone (or soil) called Turbah/Muhr for saying prayers, but the Sunni Muslims don't use that and also consider it something taboo.
Why there is such difference in style?
The only material we pray on is mud that is pure and clean. We don't pray on clay to "worship it", haply we pray on it because it is considered as a pure material, and as all Muslims — Shi'a and Sunni — believe that one must be clean and in a clean environment to pray.
I once asked my Sunni friend: "Why do you Sunni people pray with your shoes on in the mosque? Isn't the mosque an untainted and a holy place? What if you get mud on shoes, can you still pray with them?" She replied with: "Yes we can pray with our shoes if there's mud on it because mud is 'taher' (clean), so it doesn't really matter" I told her that "Us Shi'a Muslims pray on mud merely because its 'taher' (طاهر) and Sunni Muslims object that and believed that we are worshiping them as idols"
The point is that Shi'a Muslims pray on a 'turbah (تربه)' (which comes from the word tarab that means mud) because it is taher and pure. And the Prophet (pbuh) used to pray on pure flat GROUND because in the Prophet's time they did not have a sijadah as there was no technology back then to produce them.
بسم الله الرحمن الرحیم
First of all, it would be considered as a constructive and helpful issue to know that Shiite school of thought doesn’t believe that they should prostrate solely on the stone. In fact the belief of Shiite is that they should prostrate on the things which are counted as a part of the earth. Meanwhile, it can be significant to note that based on both Shia and Sunni scholars, it is permissible to prostrate over stone, clay and other similar materials.
Unfortunately, many people assume that Shias are prostrating on the stones, because, but it is not right what they're assuming. Actually the truth is something else. As I mentioned early, Shias prostrate (do Sujud) on the materials which can be considered as a part of the earth as natural materials, NOT ARTIFICIAL MATERIALS (which are handmade). Consequently, although dear Sunni brothers prostrate on the carpets, but Shia brothers believe that they mustn’t prostrate on the carpet. Since it could be considered as an artificial material which is made by humans, and it’s not a natural material, thus we must not to prostrate on the things which are considered as a handmade thing, since prostration is for Allah, not... / Hence you must pay attention what you are prostrating upon. Then, I reckon it can be a rational and logical reason for fair and impartial people (beside Shia narrations) that we should not Sujud or prostrate for Allah on artificial materials which are made by humans. On the other hand, there wasn’t such carpets at the period of Prophet Muhammad (pbuh), and he did prostrate on natural Hasirah (mat) as both Shia and Sunni agree that.
For instance, you can see some related hadiths (traditions) from Sunni about prostrating on the earth, mud (in order to support my claims. (although there are more hadiths concerning it, but I named some of them (e.g.), otherwise to see more related hadiths, inform me to present you more narrations)
Abu Hazim narrates about how the Prophet SAWW prayed:
He faced the Qiblah and the people stood behind him. He recited Qur’an, then bowed and raised his head, then he moved backwards until he prostrated on the earth, then he went back to the pulpit and recited Qur’an, then bowed and raised his head, then he moved backwards until he prostrated on the earth.” http://sunnah.com/urn/1287640
The Prophet used to place his forehead on the earth while prostrating. / Saheeh Bukhari, Part 1, p. 97
And: Abu Sa'eed used to relate that he had seen the Prophet prostrating on wet mud, so much so that he could see mud stains on his forehead. (Saheeh Bukhari, Part I, p. 104)
Therefore we ought not to accuse Shia why they prostrate on the stone (Turbah)… . As more explanation, as logical illustration for dear fair users:
Many of time it is seen that Shias prostrate on a small piece of stones(Turbahs), actually a reason in order to persuade the fair people could be related to the nowadays life. In fact, whereas it is not easy to find the earth to prostrate (because there are many artificial things like carpets and … on the homes and even mosques, then Shias use a stone or Muhr (turbah) as a small portable part of the earth to do Sujud (prostrate) on it instead of carpet or other artificial materials. Otherwise I reckon we should remove the carpets whenever we intend to prostrate… in order to prostrating on the earth! (which can be hard...)
As another helpful issue: Unfortunately some people assume that Shias are worshiping the stone not Allah! , but this is completely a wrong issue (as a negative propaganda against shia…), haply a piece of stone (or Muhr in Farsi and perhaps in Urdu or Turbah in Arabic) is considered as an easy portable piece of the earth which Shias do Sujud by that intention, not not doing Sujud for the stone or Turbah!. Otherwise Shia believe that it is permissible to prostrate at other natural things which are gotten from the earth as well. Solely as my personal opinion:
E.g. sb accuses prostrating on the stone or Turbah, s/he can accuse about prostrating on the carpet as well (as really an artificial material). Then we ought to survey many issue based on the hadiths fairly, not by paying attention to negative propagandas. [as Shia has evidences and reasons for other its acts too, then it is not fair to accuse it according to negative propaganda…, hence I strongly suggest you to ask your ambiguities about Shia from their experts, not from… / I am at your service to answer you according to my common limited knowledge)
Anyhow, as I mentioned before, it is permissible to prostrate (do Sujud) on the earh (by its conditions) based on Shia school of thought. For instance you can see the following tradition from Shia as well as a related issue to what we can prostrate on. It is narrated that Hisham b. al-Hakam said to Abu Abdillah (Imam Sadiq S.A): Inform me of what it is permitted to do sujud upon and what is not permitted. He answered:
Sujud is not allowed except for upon the earth or what the earth grows except that which is eaten or worn. So he said to him: May I be your sacrifice, what is the reason for that? He said: It is because sujud is submission to Allah, azza wa jalla, so it is not appropriate that it be upon what it eaten and worn. For the sons of the dunya are worshipers of what they wear and eat. And the prostrator in his sujud is in the worship of Allah, azza wa jalla, so it is not appropriate that he place his forehead in his sujud upon that which is worshiped by the sons of the dunya, having been deceived by its deception. And sujud upon the earth is better for it is more (conducive) to attaining humility and submission to Allah, azza wa jalla. (The reference of hadith: Wasā'il al-Shīʿa vol, 3 page 591 hadith no.1
God willing, perhaps, I’ll edit and complete my answer as a more complete answer in the future.
May Allah guide us
I will be mentioning the view of the Shia and Sunni in his answer
Why don't Sunnis pray on a stone made from clay ?
Sunni don't use a stone (Muhr) as the shia because there is no authentic hadith from the prophet (PBUH) which say we should only pray on a stone. On the contrary, Praying on Mud/Clay only contradicts the verses that claim Islam is not considered a difficult religion.
Allah intends for you ease and does not intend for you hardship,... [Quran 2/185]
He has chosen you and has not placed upon you in the religion any difficulty. [Quran 22/78]
Thus, Sunnis are allowed to pray on anything (mud, dust, stone, cloth, etc...). Not to mention it should be clean (don't pray on the bathroom floor).
The Prophet (may peace be upon him) said: I have been conferred upon five (things) which were not granted to anyone before me (and these are): Every apostle wassent particularly to his own people, whereas I have been sent to all the red and the black the spoils of war have been made lawful for me, and these were never made lawful to anyone before me, and the earth has been made sacred and pure and mosque for me, so whenever the time of prayer comes for any one of you he should pray whenever he is, and I have been supported by awe (by which the enemy is overwhelmed) from the distance (which one takes) one month to cover and I have been granted intercession. [Sahih Muslim - sunnah.com]
Narrated Maimuna: (the wife of the Prophet) During my menses, I never prayed, but used to sit on the mat beside the mosque of Allah's Apostle. He used to offer the prayer on his sheet and in prostration some of his clothes used to touch me." [Sahih Bukhary - sunnah.com]
I should also mention that there are other narrations in Sunni books such as At-Tabri, kanz ul ammul and other books which mention the prophet used to order Muslims to pray on Mud/Clay only. Sunnis don't acknowledge these narrations because these are either fabricated or really weak and because of the two authentic hadiths (quoted above) which contradict these narrations.
Why do Sunnis consider it taboo ?
Sunnis consider it taboo because it contradicts the the Quran and the hadith as I mentioned earlier.
Sunnis don't like the idea that the clay used by Shia is considered the best if it was from Kerbala. This promotes the idea that Kerbala is considered holier than Mecca or Madinah which is a really serious issue in the eyes of the Sunnis.
Why do shia pray on a stone made from clay?
One of the conditions for the validity of prayers according to shia is to pray on clean and pure earth. One shia hadith attributed to an Imam specifically orders shia to pray on clay/mud only. Source of this hadith is found in the book Wasa’il al-Shi’ah, Volume 3 page 592
Imam Ja’far al-Sadiq said: "Prostration is not permitted except on the earth and whatever grows from it except on those things that are eaten or on cotton.” When he (Imam Ja’far as Sadiq a.s) was asked whether having one’s turban touch the earth instead of the forehead was acceptable, he replied that this was not sufficient unless the forehead actually touched the earth."