"Assailant" vs "Attacker"

Besides sports in which an attacker is an offensive player, is there any difference between assailant and attacker?

a person who attacks somebody

I guess attacker can also be used for animals but not assailant. Is there anything else or are they fully interchangeable when describing a person who physically attacks somebody else?

Can both words also be used when talking about psychological attacks, like insulting, bullying, etc?

Yes, the two words are used interchangeably, especially when referring to the act of physically setting upon a person.

To trace the difference between them, we take a look at their Verbs:

"Assail" means:

to attack a person violently to cause hurt to them. The attack may be physical or verbal. When you assail a person (verbally), you abuse them. Assailing consists in hitting a person very hard and repeatedly.

"Attack" means:

to set upon with violence and applies to offensive action, especially to the onset of planned aggression. It also means to criticize strongly in a hostile manner.

Assailant comes from the same root as assault, which has connotations beyond just attack. At one time, newspapers were not allowed to use the word sexual, which led to overuse of assault in reporting rapes. There is an alleged news story (much too good to check) along the lines of 'The woman was grabbed from behind, overpowered, and left bleeding and unconscious. She was fortunate not to have been assaulted.'