Hi Tim,
As you know, both are open source, web browser-based collections management systems with similar licensing. To go over all of the functional and technical differences would make for a very long article, so for now I'll stick to the highlights (as I see them) here.
CollectionSpace is designed around the SPECTRUM data standard, a comprehensive standard for documenting and describing collections published by the UK Collections Trust. It's a fine standard for museums with conventional processes but may not be ideal for a family archive. However, there are a number of available "extensions" to the core CollectionSpace schema, including a local history one that may make a better fit for your project. You can see the list of extensions at https://wiki.collectionspace.org/display/collectionspace/Domain+Extension+Sets+and+Community+of+Practice+Profiles. These extensions are mostly just definitions of additional fields. You'll still be using SPECTRUM at the end of the day.
CollectiveAccess takes a somewhat different approach. We make the virtually all of the schema configurable. The set up of fields, editing screens, workflows and more can be customized to suit your specific project requirements. If you want a SPECTRUM-like system you can configure it in CollectiveAccess, or you can implement some other standard(s). Or you can just make it all up :-) Totally custom systems are not uncommon. Sometimes you need to go off-road.
The flipside to this flexibility in CollectiveAccess is that you have to articulate what is needed at the start. There's no specific "base" setup that you can slide into without thinking and work in forever. There is, however, a library of "installation profiles" – pre-made configurations – you can use as starting points for your own set up, as well as a few basic default setups (based upon DublinCore, and some other simple standards) that you may use to get a feel for things. With a bit of planning and upfront effort you can use CollectiveAccess to implement a system precisely adapted to the nature of your collection.
Another key difference is focus. CollectionSpace is specifically designed for museum collections. CollectiveAccess was initially designed for museums but has been expanded over the years to support archival and special collections processes. It also has some of the features of a digital asset management system, including support for a wide variety of digital media formats (video, audio, documents). If your project is both collections and archives oriented then CollectiveAccess has features that may prove very useful. If your project is purely archival you may want to consider an archives-centered system. The two most common are AccessToMemory (AtoM) and Archivespace. Both are great options for managing a conventional archive.
I hope this helps!
Seth
