There are a number of differences between archival and non-archival versions of foamcore and these differences will vary between different manufacturers.
A general guideline is that regular foamboard and regular matboard are considered ok for temporary and/or very inexpensive framing. I am not a conservator, but I have heard that minor decay in the artwork will start to occur within 5 years. This will vary dramatically based on the type of framing components, the environment (light, temperature, humidity) of the frame package, the chemistry of the print, and type of mounting. Although some decay may occur, it may not be very noticeable without doing an A/B comparison with the original.
Archival mat board and foam board will usually have the following characteristics:
Composed of acid free material
Composed of lignin-free material
Buffered to help maintain the alkaline pH
Bainbridge has a Micro Chamber technology called Artcare™ that goes a step further and allows their archival products to actively absorb harmful components entering the framing package. This can be especially useful in the case of foamboard. Archival foam board will have an archival mounting surface to protect the print, but the foam in the center will out gas over time. The Bainbridge product will absorb this out gassing.