There are some apparent differences which exist, and have existed between Shīʿas and Sunnis; for instance, the folding of arms during prayer as practiced by the Sunnis, or the method in which ‘wuḍhu’ is performed can be cited as examples of apparent differences (in the form of practices) in Jurisprudence between the two schools. The causes for these differences are actually rooted in more fundamental issues, which have been discussed in detail elsewhere on this site.
As to why Shīʿas recite ‘Alḥamdulillāh rabbi al-ʿālamīn’ instead of ‘Ameen’ after sūrah ‘al-Hamd’:
Prayer is amongst the acts of worship in which the contents (of prayer) are determined by the Holy Qur'ān and traditions from the Holy Prophet (pbuh) and his household (they are not derived through rational). In prayer, one must not recite anything other than ‘ḏikr’; otherwise the prayer would become void. Just as the Shīʿah have obtained all the rest of their teachings through the instructions and commandments of the Prophet and his household (pbut), in this matter too they act in accordance to their instructions. Based upon narrations, the Ahlulbayt have told us that ‘saying ‘Ameen’ after reciting sūrah ‘al-Hamd’ is not part of the prayer, hence it is not permissible to say it’. It is narrated from Imam Sadiq (pbuh): ‘Whenever you are praying in congregation, and the leader of the congregation finishes reciting ‘Al-Hamd’, do not say ‘Ameen’, say instead ‘Alḥamdulillāh rabbi al-ʿālamīn’.
The maraji’ have given the following fatwa based on the mentioned narration: ‘Of the acts which invalidate the prayer; is saying ‘Ameen’ after reciting ‘al-Hamd’, although if someone says it mistakenly or out of ‘taqīyah’ (dissimulation), the prayer is not void’.
Finally, it is necessary to mention that according to Shīʿah Islamic law, it is not actually compulsory to say ‘Alḥamdulillāh rabbi al-ʿālamīn’ after ‘al-Hamd’; it is just mustahabb (preferable). However, most Sunni schools of thought consider it an obligatory part of prayer, hence the onus is upon them to verify their claim through the Qur'ān and Prophetic Traditions.
 Tusi, Muhammad bin al-Hasan, al-Tahdhib, vol. 2, pg. 74.
 Tawdih al-Masa’il (annotated by Imam Khomeini), vol. 1, pg. 617. This ruling is agreed upon by all the respected jurists.
 Ibid, pg. 559.
 Of course, according to the explicit mention of great Sunni figures and commentators of the Quran, saying ‘Ameen’ is not part of the prayer. (Muhammad Rashid Ridha, al-Minar, vol. 1, pg. 39)