Compare Sunbeam's opening line:
"A beam of brilliant light flashes out from your hand in a
5-foot-wide, 60-foot-line."
to Sunburst's opening line:
"Brilliant sunlight flashes in a 60-foot radius centered on a
point you choose within range."
This seems to suggest that, for the purposes of Sunlight Hypersensitivity (e.g., that of Vampires), Sunbeam and Sunburst are different. However, Sunbeam has a second component:
"For the duration, a mote of brilliant radiance shines in your hand.
It sheds bright light in a 30-foot radius and dim light for an
additional 30 feet. The light is sunlight."
Does the sunlight "property" apply only to the mote, or to the beam attack as well? RAW, it would seem like the former (based on the quotes above; the name of the spell suggests otherwise), but it's counterintuitive that the least powerful component of a spell (non-damage dealing, presumably 'utility-oriented') has a damage-dealing property that the more powerful, damage-dealing one doesn't.
One other wrinkle:
RAW, it would seem that Sunburst doesn't trigger SHS, despite the fact that it's a L8 spell that explicitly creates sunlight. Is it reasonable to house-rule that the affected vampire(s) takes the 20 damage, at the time of casting (even if they succeed the saving throw, since the entire burst is still sunlight)?
Best Answer
As is noted in your answer, the mechanics here are pretty clear:
The vampire takes normal damage from Sunbeam (6d8, with the option to repeat on subsequent turns) and Sunburst (12d6).
In the case of Sunbeam, the vampire takes an additional 20 damage at the start of its turn, if within the 60-foot radius.
The vampire doesn't take extra damage from Sunburst, because there is no sunlight at the start of its turn. Simple enough.
Which is "better" depends on the situation. 12d6 will generally do quite a lot more damage than 6d8, but 20 damage / round narrows the gap almost immediately.
This does mean that in this particular instance, the level six spell may be a bit better suited to the situation than the level eight spell. Never fear, the eighth level spell will usually be a lot better (hits more targets for more damage, does other useful things).
Mechanics vs. Flavor
The mechanics above are pretty clear. But the mechanics may not fit the lore you have in your head about vampires (I think this is what SSD is getting at with his comment). As the DM, you have two options here.
Fit the lore to the mechanics
The first is to fit the lore to the mechanics. Here, it's pretty simple... Vampires burn in the sunlight, but it takes a little sizzling before they start to hurt. Sunburst just doesn't shine the light on them long enough to cause problems (other than, you know, the massive amount of radiant damage and potential blindness).
A longer dose of sunlight gradually sets them on fire, triggering their weakness.
Bend the mechanics to fit (your version) of the lore
Sunlight hurts vampires, right? So go ahead and throw an extra 20 damage onto Sunburst. It's not the rules, but if it makes the fight more fun, then that can be fine.
Which you choose will depend on your group, and how often you bend things in their favor ("hey, 20 extra damage!"), vs. how often you bend things against them.