Congratulations to you and your bride, Andrew! I'll just join the free-for-all for a minute.....
Some posters have contrasted Vancouver with Banff as water vs. mountains. In fact Vancouver is ringed on three sides with mountains, maybe not the Canadian Rockies but still spectacular. Some of them right in your face as you look north from the city.
I don't know what snow conditions are like in the Banff-Jasper area in early June, but they might inhibit hiking beyond the towns. Worth checking into anyway. No such problem with Vancouver and its lowland surroundings. I'm assuming that you don't expect to ski in June.
Jasper is smaller and more scenic than Banff, I think, so might be a better choice in the Rockies. Not far up the very scenic Icefield Parkway from Banff. But see above re possible snow issues in June (not the roads, the trails).
Not really my business, it's your honeymoon, but I'd pick one or the other area instead of trying to split between Vancouver and the Rockies. And, after all, neither is hard to get to in the future.