Hello,
unfortunately, the mix of company, product and library names can be quite confusing to figure out if you are not already familiar with them.
<ul>
<li>Alfresco - the name of the company and often / for historical reasons also used as the name of the primary ECM product, which is available as Alfresco Community (open source) and Alfresco One (commercial)</li>
<li>Activiti - the name of the BPM open source library that is both embedded into the Alfresco ECM product (Alfresco Community and Alfresco One), but also exists as a separate, commercial product called "Alfresco Activiti"</li>
</ul>
So your question might be if you need Alfresco Community / One which has Activiti embedded, or Alfresco Activiti, which is Activiti without any document management, but provides a lot of corporate-level business process management features that Alfresco Community / One do not have. E.g. allowing business users to design, model and activate business processes in the live instance, adding process analytics etc.
For developing document / content-centric business processes, the Activiti engine embedded in Alfresco Community / One is perfectly capable of supporting that and then some. But if you want to "only" focus on business process management and integrate some existing / custom applications with a central process management service, without actually having a strong focus on documents / content, then Alfresco Activiti is typically what you'd want.
Of course the two can also be used together and integrated with one another throught the (commercial) Alfresco Activiti Integration addon.
Regards
Axel