What (if any) are the differences between [The Standard for Software Component Testing](http://www.testingstandards.co.uk/Component%20Testing.pdf) (from the British Computing Society SIGIST and available for free) and BS 7925-2 (which is available to purchase for £170)?
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The first document appears to be the last draft version of the published (expensive) version - in fact, this page states that and appears (to my reading) to be recommending that you use the draft: <http://www.testingstandards.co.uk/bs_7925-2.htm> 

What reasons do you have for using either standard at all? 

Neither will help you to fulfil your stakeholders' needs for relevant, timely, quality-related information - only you can do that, by doing a good job of discussing their needs with them, analysing the testing problems that you have, and staying in close communication with them during the course of the project.

If your stakeholders have specific needs for following a particular standard, then you need to explain the context for that before anybody can help you.
You probably better spend 164 CHF and purchase the draft ISO 29119 parts 2 & 3 (parts 1 & 4 are still not for sale), it is partially based on BS 7925-2 and might replace it in the future.
I'm reading it now, and it seems they did a good job with it, it gives you the right foundation but leaves a lot of context to be added.