When we start talking about prayer, we should to be aware of which practices within that prayer are wajib (compulsory) and which are mustahabb (recommended or optional).
Before the actual prayer starts, it is common but optional to read the adhan and iqaamat.
There are differences between the Sunni and Shia adhan as detailed in this wikipedia entry.
Shia add the line "Ash hado anna Aliyan wali-Allah" in their adhan, however they all unanimously agree that this line is not part of the adhan and is optional.
With regards to the actual prayer:
- Shia pray with their hands by their sides, whereas Sunnis either pray with their hands folded, or by their sides depending on the madhhab (Maliki's from West Africa pray with their hands by their sides).
- After the rukoo, when standing back up again, it is common, but optional for Shia to say "Samey Allahho-layman hamedah" (Allah listens to one who praises him).
Sunnis also say this - In between the two prostrations, it is common but optional for Shia to say "Astaghfir-oollaha Rabbee wa atoobo-ilaih".
- After the 2nd prostration when standing back up again, after saying Allahu-Akbar, Shia say "Bay hawlillahay wa quoowa-teyhe aqoomo aqood" (Due to the vigor given by Allah and because of the vitality from Him, I rise and stand).
- In the second rakat, after the second surah, it is very common, but again optional for Shia to read a dua/qunoot. While Sunnis also believe that qunoot is an optional part of prayer, from what I've seen it is rare.
- In a 3 or 4 rakat prayer, after the 2nd prostration of the 2nd rakat, Shias only recite the tashahhud.
I believe Sunnis recite both the taslim/salaam as well as the tashahhud. - After the last rakat and final prostration, both Shias and Sunnis recite the tashahhud and taslim/salaam which officially ends the prayer, immediately after which, Shias optionally say Allahu-Akbar 3 times and recite salawat 3 times.
- Immediately after reciting tashahhud (in either 2nd or last rakat), it is wajib for Shia to recite a salawat.
- I believe there are some differences in the Sunni and Shia taslim.
- Shias prostrate on to a small clay or stone block - for why, see this questionsee this question.
Some notes:
- The salawat after the tashahhud is the usual salawat to the Prophet (ص) that all Muslims read, i.e.: "Allaahumma salli `alaa Muhammadin wa Aali Muhammad". It is incorrect to say that this is meant just for Imam Ali (ع).
- Once the taslim has been read the prayer is finished. Anything after the taslim are optional mustahabb pratices.
- One of these optional practices is to read a tasbeeh, i.e. 100 beads on a string - 34 x Allahu Akbar, 33 x Alhamdo lillah, 33 x Subhan-Allah
- While a very tiny minority of Shia add the line "And I bear witness that the Amir of the Mu'mineen is Ali, the wali of Allah" to the tashahhud, this is due to ignorance. All Shia scholars and jurists unanimously agree that adding this line is bidah and will make the prayer void.