The difference is that ArchivesSpace is an 'archives information management system', whereas Alfresco is a full 'content management system', which means that it can manage any type of content.
What ArchivesSpace is:
ArchivesSpace Version 1.0 was completed in August 2013. It includes basic functionality for accessioning, processing, description, digital object description, and authority control workflows for archival material, as well as for searching descriptions and exporting metadata objects such as EAD, MARCXML, MODS, Dublin Core, METS, and CSV.
http://www.archivesspace.org/developmentplan
As for Alfresco:
The Alfresco One platform allows organizations to fully manage any type of content from simple office documents to scanned images, photographs, engineering drawings and even large video files.
http://www.alfresco.com/products/one/aws?utm_expid=11184972-12.IcCW-3j6RMavigPGfjODyw.1&utm_referrer=http%3A%2F%2Fwww.alfresco.com%2F
What the difference ultimately comes down to is not what it can store but what functionality you get in addition. ArchivesSpace seems to be a simple implementation of a document storage system that stores documents in collections with associated metadata. Alfresco also offers workflows, custom actions, previews, sites, wikis etc.
If your specific use case is related to archiving off documents specifically and you want something that will already be good at this then go ahead and use ArchivesSpace, if not, or if you want to expand the system out in future, then Alfresco will likely be able to do more but will likely take more effort to configure to your specific use case as you will have to create a custom content model and such.
Alfresco Records Management is for managing documents that will likely have some legal significance, such as court papers, official government department responses etc, and as such their creation and destruction need to be closely managed. As far as I can see this is not something ArchivesSpace can do.
(Full disclosure: I work for an Alfresco partner)