An activity is a sub-set of process. For example, new business process in Insurance industry can comprise of activities such as Application Data Entry, Underwriting, and Policy Issuance.
Whereas an event is an occurrence or an outcome which is of significance, and based on which typically, a business rule can be triggered. For a example, an event could be start of an activity, end of an activity, non completion of an activity within a certain timeframe, etc.
A workflow engine, can trigger an escalation rule, if Underwriting activity is not completed by an underwriter within 2 hours of its assignment. In this case 2 hours of non completion of Underwriting activity is an event.
A workflow engine can send an sms or an e-mail on issuance of an insurance policy to the customer. In this case, issuance of the policy, which is completion of issuance activity is an event.