The scenario you present, compared to the RAW and RAI(tweet), does not yield two spells cast at 4th level (power) even though a 4th level spell slot was used.
Result: The 3rd level Daylight spell will overcome the 2d level Darkness spell, per spell description on PHB 231.
You can burn a higher level spell slot for either spell, but Darkness' or Daylight's "power" is still 2d or 3rd, respectively.
From Chapter 10, PHB, Spellcasting, under Spell Slots
When a character casts a spell, he or she expends a slot of that spell’s level or higher, effectively “filling” a slot with the spell. You can think of a spell slot as a groove of a certain size—small for a 1st-level slot, larger for a spell of higher level. A 1st-level spell fits into a slot of any size, but a 9th-level spell fits only in a 9th-level slot.
The rules do not say that all spells have more powerful effects when cast at a higher level. The rules do specify what the more powerful effects of some spells are "at higher level" in the text of those specific spells. This answer addresses a similar question.
Point 1. Specific versus general. Does the general spell casting rule overcome the specific spell description? (See Provisional Conclusion at the end).
Point 2. Do all spells accrue more power when cast with a higher level slot? No. Mr. Crawford's response indicates that they do not. (See Provisional Conclusion at the end).
Look at the specific spell in the book and find the "at higher levels" description (or lack of it). It is usually at the end of the spell description.
Example from Fireball (Basic Rules Set page 90), shows the end of the spell description as:
"**At Higher Levels**. When you cast this spell using a spell slot of 4th level or higher, the damage increases by 1d6 for each slot level above 3rd."
The last sentence from the Darkness spell (Basic Rules, page 87; PHB page 230 matches Basic):
"If any of this spell’s area overlaps with an area of light created by a spell of 2nd level or lower, the spell that created the light is dispelled."
If any of this spell’s area overlaps with an area of darkness created by a spell of 3rd level or lower, the spell that created the darkness is dispelled
Daylight also lacks any "at higher level" language. PHB page 231.
Since the Darkness spell does not include "at higher level," the lack of RAW for that feature matches Mr. Crawford's tweet:
As written, the darkness spell can't be made more powerful with a higher level slot.
The same can be logically applied as true for the Daylight spell.
Why This Was a Good Question in the First Place
Getting more power with more spell slots, example of Dispel Magic (3rd level spell, p 88 Basic Rules).
Any spell of 3rd level or lower on the target ends. For each spell of 4th level or higher on the target, make an ability check using your spell casting ability. The DC equals 10 + the spell’s level. On a successful check, the spell ends.
At Higher Levels. When you cast this spell using a spell slot of 4th level or higher, you automatically end the effects of a spell on the target if the spell’s level is equal to or less than the level of the spell slot you used.
Since in some cases light or dark can overcome (dispel?) dark or light, it was not unreasonable to try to apply the same reasoning as shown in the dispel magic spell, since it is a "spell versus spell" case that was presented.
Full text from Casting Spell at a Higher Level: (Basic Rules, P 78).
When a spellcaster casts a spell using a slot that is of a higher level than the spell, the spell assumes the higher level for that casting. For instance, if Umara casts magic missile using one of her 2nd-level slots, that magic missile is 2nd level. Effectively, the spell expands to fill the slot it is put into.
**Some spells**, such as magic missile and cure wounds, **have more powerful effects when cast at a higher level, as detailed in a spell’s description.**
The "some spells" ... "have more powerful effects" directs us to the specifics of a given spell to find out if there is more power in the spell when cast at a higher level. It is from the lack of saying "all spells ... have more powerful effects" that leads me to that conclusion. Without that spell description, "at higher levels," the spell cast with a higher slot has the higher level but not higher effects / power.
I admit that it took the input from Mr Crawford to sell me on that. It is fair to say that this response relies on dev intent as much as RAW.