Why did the Avro Vulcan use a delta wing?
It was obvious to the design team that conventional aircraft could not satisfy the required specification; knowing little about high-speed flight and unable to glean much from the Royal Aircraft Establishment or the US, the team at Avro investigated German Second World War swept wing research.
The team estimated that an otherwise conventional aircraft, with a swept wing of 45°, would have doubled the weight requirement. Realizing that swept wings increase longitudinal stability, the team deleted the tail (empennage) and the supporting fuselage, it thus became a swept-back flying wing with only a rudimentary forward fuselage and a fin (vertical stabilizer) at each wingtip.
The estimated weight was now only 50% over the requirement; a delta shape resulted from reducing the wingspan and maintaining the wing area by filling in the space between the wingtips, which enabled the specification to be met.
— Wikipedia

In short, the team didn't know much about high-speed flight (blame Wikipedia). And to reduce the weight, they eliminated the tail-plane, shortened the span, and filled in the space between the wingtips. Achieving their weight target.
For the same area, a wide wing is lighter to attach, as it requires smaller spread spars.
So, for weight reduction, a delta wing wins against a conventional swept-back wing aircraft. In all, great engineering solution to a big problem, with one downside—
—higher fuel burn at those speeds, limiting the range.
The B-52's were known for loitering in the air for extended durations. Their efficient design requires fewer rendezvous' with the tanker. Unlike the famous Vulcan train.
Some numbers (in kg and km):
        empty weight     MTOW      payload+fuel      range
B-52      83,250        220,000      136,750         7,210
Vulcan    37,144         77,111       39,967         4,171


Maintaining the same wing area; saving weight by reducing the heavy structural elements—the tail-plane, big spars, and wing-span. (Eyeballed the scale, only illustrative.)