When I pray salah the mosque, some people ( 2-3) pray a bit differently. They raise their hands in most of the takbeers even though the rest of us only raise our hands during takbeer tahrima. They also raise the fingers continuously up and down in tashahud whereas we just do it once when the name of Allah is said.
Muslims follow in their prayer the way the Prophet () used to pray, Muslim scholars have different opinions on what really is a reliable hadith and what is not, based on that they defined what is a sunnah of the Prophet and what isn't, that's the reason why some Muslims pray differently then others.
For evidences for your two addressed issues, see for example these narrations from sahih Muslim and sahih al-Bukhari and al-Muwatta' which are evidences for raising hands for almost all takbeers.
While the "opposite" opinion which is raising the hands only once is based on ahadith -in first place- narrated on the authority of ibn Masu'd like in sunan abi Dawod, jami' at-Tirmdihi (who said this is the opinion of some sahaba, and the tabi'yn, Sufyan a-Thawri and the scholars of al-Kufa) and sunan an-Nasa-i.
And this hadith for raising the fingers tashahud (from sunan an-Nasa-i)
... Then he sat up and placed his left leg under him; he put his left hand on his left thigh and knee, and he put the edge of his right elbow on his right thigh, then he held two of his fingers together and made a circle, and raised his forefinger, and I saw him moving it and supplicating with it.
For further information also take a look at posts showing some differences and their evidences like in: