I agree with Jeroen that combining the 2 would make a very nice trip.
The Tour du Mont Blanc (TMB) is one of the most popular trips in Europa so walking alone isn't a guarantee. But the views are fantastic. Don't be mistaken, you don't walk around a single mountain but around a mountain range with Mont Blanc on top but with a number of other mountains which have earned their place in (European) mountaineering history: Grand Jorasse, Dru, ... From the NW (French)side you can almost look into the heart of the range, aspecially if you're watching from the so-called southern balcony above Chamonix. An impressive view on glaciers and snow packed mountains. On the SW (Italian) side, the views are very different with the walls appearing to rise nearly vertically. Above Courmayeur, near the refuge of Le Pré is a lovely spot for camping with a fantastic view to the south face of the Grand Jorasse. A very nice spor to wake up in the morning.
As already said, it's a very popular area, supported by a good hut system so it can be hard at times to find a spot for your own, but if you can take the variations of the normal route, it could be a bit easier. BTW, I prefer the variations above the normal route, more challenging but nicer also.
I don't know the Haute Route as well als the TMB, although I have an idea about the route. There are very nice parts but as Jeroen also noted some paved tracks which I would prefer to avoid.
You could combine both and perhaps even change the route here and there to make it even better. At least the first part of the HR in Switzerland (the Franch part of the HR uses the same paths as the TMB) is something that seems less interesting and personnaly I'm thinking of circumventing this part by following the Tour du Grand Combin in the anticlockwise direction, entering first into Italy and then crossing the border into Switzerland and passing the cabane/refuge the Chanrion. I haven't figured out al the details but you should have some fantastic sights on the glaciers of Swiss Valais/Wallis.
Ofcourse there are other options overthere, Tour de Matterhorn, Tour de Monte Rosa (although I believe you have some short glacier traverses on those tours)
Closer to Zurich, you have the Berner Overland with famous mountains like the Eiger and the biggest glacier in Europe, the Aletch, ...