Vancouver is amazing! I've lived both there and in Victoria (where I currently live) and while Vancouver Island is indeed a special place, I miss Vancouver. Partly because I have close friends there, but also because it's a bigger, more diverse urban centre than Victoria.
The natural beauty in all these areas is unsurpassed, but Vancouver has the coast mountains -- closer than the Cascades are to Victoria, so a more important presence. The parks, beaches, neighbourhoods for walking -- all are terrific. The food is fantastic! Shopping is decent -- for a visitor from the US, prices will be good (since you get more for your $ here).
One week isn't long, though, so you might prefer to stick to two spots. Unless you will be flying to the Island, it takes an hour and a half by ferry to get here. (GORGEOUS ferry ride.-- well worth it! But the flights are faster, of course.) If you are taking the ferry, you should reserve unless coming on foot -- because in summer they get very booked up.
It really depends on your interests and what type of a holiday you are after, though. Each of these places has fantastic hiking, biking or mountain biking, water sports, scenery. Vancouver has more in the way of restaurants, theatre, music, museums, shopping.
You can PM me with specific questions! And have fun planning!
