Your current hit points may never exceed your hit point maximum. Unless a rule specifically states it also affects hit points (not just hit point maximum), only the hit point maximum is lowered.
If your hit point maximum drops to below your current hit points, your current hit points will be reduced to the new hit point maximum.
Keep in mind that certain effects cause both your hit points and hit point maximum to be reduced. An example is the attack of a Wraith which causes damage and then lowers the characters hit point maximum if the save is failed.
Example:
A level 3 Barbarian with 14 Constitution and 39 hit point maximum with 31 current hit points loses 1 constitution due to disease. His hit point maximum drops to 36, but his current hit points stay at 31.
The disease is horrible, and even before healing he loses another 4 constitution (total of 9 now). His hit point maximum drops to 30, and, since his hit point maximum is now lower than his current hit points, his current hit points drop to 30 as well.
No, otherwise it would be mentioned by the spells' descriptions.
That being said, I believe the reason why it's not mentioned are the spells' intended usages / their themes.
Lesser Restoration allows you to remove the following conditions:
disease, [...] blinded, deafened, paralyzed, or poisoned
All of them are physical conditions, leading me to believe that the intent of the spell is to be your all-rounder spell for minor illnesses etc. Frightened is not a physical condition, therefore it doesn't fit the theme.
Moving on to Greater Restoration, it can remove the following effects:
- One level of exhaustion
- One effect that charmed or petrified the target
- One curse, including the target’s attunement to a cursed magic item
- Any reduction to one of the target’s ability scores
- One effect reducing the target’s hit point maximum
It removes major detrimental effects, and once again, they are all physical effects (and curses), aside from the Charmed condition. In my opinion, it would be reasonable to include Frightened alongside Charmed, but since Charmed can be way more impactful than being Frightened (in one case, you run away, in the other, you might fight for the opposing team), that's probably why they chose Charmed instead of Frightened.
In addition to the previous paragraphs, there are spells that do help against the Frightened condition (sometimes limited to against specific types), just not the Restoration spells:
- Protection from Evil and Good: immune against being Frightened (and other effects) by aberrations, celestials, elementals, fey, fiends, and undead, and advantage on saving throws for existing effects.
- Aura of Purity: advantage on saves against Frightened (and other effects)
- Heroism: immunity against the Frightened condition
- Heroes Feast: immunity against the Frightened condition
- Calm Emotions: can suppress the Frightened condition for its duration, i.e. up to 1 minute. Effect resumes afterwards, unless the duration has run out.
There are probably a few others that I failed to find.
Best Answer
Current hit points are unaffected by hit point maximum increases
A character's hit point maximum is only the upper limit on the creature's current hit points:
This is reinforced in the section on healing:
Things that increase your hit point maximum (or remove an effect that was reducing your hit point maximum) have no effect on your current hit points unless stated otherwise.
This is reinforced by the wording of the aid spell:
Note that aid separately increases your max HP and your current HP. If increasing your max HP automatically increased your current HP as well, it would be redundant and potentially confusing to state both parts separately. However, because your max HP is just an upper limit, and increasing it doesn't increase current HP automatically, it's necessary to state in the spell description that both are increased.
In short: max HP increases don't change your current HP unless stated otherwise (as aid's description does).
(Note that hit point maximum decreases can decrease your current hit points, because your current HP can never be above your max HP.)