After some experimentation, and expanding on information in the comments:

Short answer

1. Because of bug.
2. But it doesn't really matter.

Long answer

[The problem]

This happens consistently in my Sanctuary settlement too, the only settlement where I have spotlights installed. I have two spotlight turrets; each is on level ground, and each is right next to its own dedicated generator.

The problem appears to be triggered by the passage of time rather than fast travel itself. For example, a very short fast travel (e.g., to Vault 111 and back) leaves the spotlights on, but a little further (e.g., to Starlight Drive In and back) and they'll turn off.

But most importantly, the problem is only cosmetic.

[The experiment]

Wait until a spotlight turret is in its darkened state, then:

1. Note that it's still moving; this strongly suggests that the game still considers it to be active.
2. Go into Workshop mode and notice your settlement's defense number.
3. Pick the turret up and set it back down. It should be lit now.
4. Notice your settlement's defense number is the same whether or not the turret is lit, so long as it's moving.

Then, to be extra sure:

1. Fast travel a good distance, for example, from Sanctuary to Starlight Drive In.
2. Return to your base. The spotlight should be dark.
3. Check your settlement's defense number. It's the same.

[So...]

While a little lighting in your settlement would be convenient—not to mention darned pretty—this appears to be a purely cosmetic bug.

You can sleep soundly knowing that those turrets, even when dark, are scaring away bad guys. And you can also sleep soundly because those turrets won't keep shining in your window. (Until Bethesda fixes this bug.)

Your ability to sneak is affected by 8 different factors:

• Equipment weight: Although you said there was no difference, there is a difference if you walk. The heavier the armor or weapon, the further you'll be heard if you move. However, if you stand still, your armor weight won't affect you at all.
• Light sources: Light sources also influence your ability to sneak heavily. However, this only affects you if an enemy has a clear LoS on you. If you are in a lit room with no enemy being able to see inside, the light does not affect you at all.
• Your agility skill: The more agility you have, the more difficult you become to spot when sneaking. This affects you in both moving and not moving.
• Perception of the enemy: This is also a huge factor which both affects you moving or not moving.
• Distance: Obviously, the further the enemy is away, the hard it will be for him to spot you.
• Any form of contact: If you touch an enemy, he will immediately spot you.
• Loud Weapons: Any form of attacking an enemy will make them spot you. However, silenced pistols, melee weapons and grenades/mines will not break your stealth, although it will cause the caution alert if you do not kill them in one hit. Grenades always cause caution alert if there are any alive enemies in the area.
• Special gear: There are several modifications like shadowed which can improve your ability to sneak. Also, gear obtained from legendary enemies can have a stealth-improving effect on you.

Your pip-boy (includes the torchlight) do not affect stealth at all as you already found out. You can also listen to radio while sneaking and nobody will notice.