I have both of their younger (cheaper) brothers in setups - unfortunately the GC here isn't very good for higher-end monitors, so no chance to audition their nicer offerings. However, similar transducers and implementations on both - just better output and electronics on the ones you're considering. So take this for what it is... a subjective review worth about $0.01.
The tweeters in the Adams are a little faster sounding and I prefer it when watching movies. It is definitely more directional than the silk domes on the JBLs but with LCR configuration that's less of an issue as you have center fill. I do prefer the JBL's for music however - especially when outside of the main listening position (which is why I moved them to the office where it's 95% music duty). Most of my problems with the JBLs for movies wouldn't necessarily apply to your use-case as you're considering the 708's which won't struggle to reach good levels in a larger room like my 305's do.
@LeftCoastTim Lots of titanium or beryllium (and several other metals) tweeters exist - metallic tweeters are somewhat common actually. AFAIK the S-ART tweeters in the Adams are a folded-foil of metallic composition as well.
I'd say you really need to audition both and see what you think (if that's possible for you, wasn't for me). Otherwise, I'd say based on the specs and reputation of both companies/models - either one is likely to give you a level of sound quality that should keep you grinning for years to come.