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Do you own both these items already, OP? Or are you going to buy a stove and pot for this trip?
If all you want to do is boil water for one person, Esbit will be marginally lighter, and somewhat cheaper if you're buying the whole system from scratch. But I would favor the canister stove, because it is more versatile, boils faster, does better in wind, and doesn't stink (I don't really mind Esbit smell) or leave residue on your pot. I personally don't care for jetboil 'cause it's a one-trick pony. Fuel-efficient and fast, yes, but kind of bulky and heavy, and for one person for one week, the fuel savings vs. weight probably won't be justified.
Smallest, lightest, most efficient pot will be something like pic related. 600-700ml titanium with lid (for lighter, this could be aluminum foil or flashing). Weight: 3-5 oz. No matter what system you use, take along a windscreen made out of foil or flashing. Surround the Esbit completely; but do not surround the canister completely, and check the canister occasionally to make sure it's not overheating.
For your trip, I would choose canister over Esbit, despite the marginally higher weight. Especially if I were buying my stove system new. I like the versatility, speed, and convenience of a canister enough to justify the extra few ounces over Esbit. Maybe this changes if you're trying to set some sort of land-speed record, but if that's the case, you might reconsider those "crap solutions" you inb4ed.
tl;dr Canister is marginally heavier, but more convenient and versatile for future trips. Esbit has its uses. Jetboil doesn't do it for me.
