'bespeel' sounds more intense to me, with more emphasis on the instrument.
Also, it's 'ik speel piano', but 'ik bespeel DE piano'
Can I also say "ik speel piano", or must I use "bespeel"?
"Ik speel piano" is also correct, and actually much more common than "bespeel".
Is there any difference in shades of meaning?
Yes, you can use 'speel' for all meanings of 'to play' and 'bespeel' only for instruments. But 'bespelen' is not often used in normal speech. In formal situations maybe more...
Why is bespeel only for instruments? What's the significance of the "be..." prefix in this situation?