I'd say it's a tough call; The Ashes will garner more coverage because it's on terrestrial telly (for maybe the last time ever) and there'll be (probably) 20+ days of sporting action, but The Lions tours are pretty special if you're a Rugby Union fan (& I am).
The Lions only tour every 4 years, so visit one of the SH giants only every 12 years. That, in a way, makes them special events. For 99% of current All Blacks this will be their one & only chance to take on The Lions. For The Lions? Well it's their shot at immortality. Only once have they won a series in NZ; that's a lot of history to overcome.
I know without looking it up that The Lions won in 1971 2-1 with the 4th test drawn, I'm afraid I'd have to look up The 1971 Ashes result. So I guess there's your answer: The Lions tour for me.