What is the difference between customs and acts of worship, and is it a condition for customs, as it is for acts of worship, that they be free from showing off and performed with sincerity?

Worship (ʿibāda) is a comprehensive term for everything that Allah loves and approves of, including inner and outer sayings and deeds. The default rule concerning it is prohibition (al-manʿ) and unlawfulness (al-taḥrīm) until there is evidence of its permissibility. As for customs (ʿādāt), they are what people are accustomed to in their daily lives, and the default rule concerning them is permissibility (al-ḥill) unless there is evidence to prohibit them. Worship requires intention (niyya) and can be tainted by showing off (riyāʾ), unlike customs which do not require intention and are not affected by showing off. Showing off is specific to acts of worship and righteous deeds. Therefore, if someone performs a worldly action to gain people's praise, this is not considered showing off.