Why do people at the mosque pray slightly differently?

One prayer, several valid ways

The Prophet ﷺ said, "Pray as you have seen me praying" (Sahih al-Bukhari 631). The Companions watched him for years, in different places and situations, and passed on what they saw. Sometimes he did a thing one way, sometimes another, and different Companions taught different cities. Later scholars collected these reports, weighed them, and formed the schools of law (madhhabs). That's why a mosque in London may have a Somali brother, a Pakistani brother and a Moroccan brother standing side by side and doing small things differently. See Do I have to follow a madhab after becoming Muslim?

Differences you might notice

• Hands while standing: most fold the right hand over the left (Sahih al-Bukhari 740), on the chest, below it, or below the navel depending on the school. Many Malikis let their arms hang by their sides.
• Raising the hands: Shafi'is and Hanbalis raise them before and after bowing, as Ibn Umar described (Sahih al-Bukhari 735). Hanafis and Malikis usually raise them only at the opening takbir.
• "Ameen" out loud or quietly after al-Fatiha in the loud prayers.
• Moving the finger during the tashahhud, or just pointing it.
• How you sit in the last tashahhud.

Why this isn't a problem

These are details within the sunnah, not differences in the faith itself. During the Prophet's ﷺ own life, Companions understood one of his instructions in two ways on the way to Banu Qurayzah, and he didn't blame either group (Sahih al-Bukhari 946). Scholars have prayed behind each other for centuries despite these differences. The distinction between the fixed revelation and scholars' understanding of it is explained in What is the difference between sharia and fiqh?

What should you do?

Learn one clear way to pray from a teacher or a reliable guide and stick with it, so you don't get confused. If someone at the mosque corrects a detail that's actually valid in another school, just smile and thank them. You don't need to copy the person next to you, and you don't need to correct them either. These differences are also unrelated to the Sunni and Shia split. See Do I have to choose Sunni or Shia to become Muslim?