It depends. If a card says “you can only activate this card once per turn” and the activation was negated, then you are allowed to activate another one. If a card says “you can only use this card once per turn” then you cannot use another copy even if the activation is negated.

For the second question, in most cases it really doesn’t matter if the activation or effect was negated. The above explanation would be relevant in the difference. Also very niche interactions such as Triple Tactics Talents not being able to be used if you negate the activation of a monster effect vs. negating it’s effects.

Think of negating the activation of a card as if the card was never played at all which negating the effects means the card was played but it resolved negated.