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.
No
An illusion of the sun created by Major Image would not be enough to impose disadvantage. It is not the brightness of the light that matters, but its source.
Sunlight sensitivity
While in sunlight, the drow has disadvantage on Attack rolls, as well as on Wisdom (Perception) checks that rely on sight.
Sunlight Hypersensitivity
The vampire takes 20 radiant damage when it starts its turn in
sunlight. While in sunlight, it has disadvantage on Attack rolls and
Ability Checks.
The sensitivity only comes into play when you are actually in sunlight. An illusion of the sun is not considered sunlight. Sunlight can, by definition, only come from the sun (or from a spell that specifically says it is sunlight).
Jeremy Crawford, lead rules designer, has agreed with this:
Q:Underdark Sunlight sensitivity. Does bright light from torches,
light cantrip etc cause the disadvantage or only sunlight?
A: The text of Sunlight Sensitivity specifies that sunlight is the
problem for the creature. Bright light in general is fine.
In other words, no matter how bright your illusion is, it is still not sunlight and still will not cause disadvantage.
Best Answer
The Daylight spell, unfortunately, does not count as Sunlight. In each spells description, it will actually mention if the spell counts as Sunlight or not, for the effects of creatures weak against it.
Here are two examples of a few spells that ARE Sunlight from page 279 of the Players Handbook (PHB).
Those are just a few I found. I believe one more exists, but I can't locate it. Note that there is also Moonbeam that adversely affects shapechangers (Which Vampires also count as) and counts as Moonlight.