The slow spell has a particularly nasty effect against spellcasters: if they try to cast a spell with a casting time of 1 action, there is 50% chance that it got delayed until the next turn (and you must spend another turn to complete it).
Then it is in the spellcaster's best interest, when they recognize that they are under the effect of slow, to avoid casting spell until it wears off (or only use spells with a casting time of 1 bonus action).
Slow, as the name suggests (although not explicitly stated), seems to imply that it causes a perceptible effect to the affected creature(s) that the entire environment is "hasted" - not just one or two creatures around them - thus allowing the victim to recognize he is under the effect of slow.
Does the victim know that they are under the effect of slow?