If the mayor is the chair of a Town and Parish Council, he or she differs form other cllrs in the following respects:
Has the right to unilaterally call and extraordinary or extraordinary general meeting of the Council
Chairs full meetings of the Council and signs copies of the approved minutes of those meetings.
Acts as a counter signatory of certain documents, e.g the Annual Governance and return
By convention, represents the Council at external meetings and events, but other Cllrs can also do that.
That;s it: The Mayor/Chair cannot for example :
Instruct other cllrs
Act in a unilateral judicial or disciplinary capacity against other cllrs
Personally authorise expenditure etc etc etc