IMHO opinion a fief is anything that was actually made into a fief. A vassal state is a state that is either in a feudal relationship to a more powerful one, and thus is literally a fief, or else a state that is in a relationship to a larger one that is simular to a feudal vassal status.
A fief was anything that was made into a fief by a feudal relationship. A fief was typically a piece of land, but didn't necessarily have to be a piece of land. A fief was held in return for a service to the lord, typically a specified amount of service in war, but could be any service.
For example, when the Ottomans conquered Rhodes, the Knights Hospitalers were now homeless. The King of Sicily (Emperor Charles V) granted them the Islands of Malta as a fief, on the terms of paying one falcon per year. The novel and movies The Maltese Falcon claim that for the first year's payment the knights sent not a living falcon but a golden bejeweled falcon.
A typical bottom level fief was a manor, a large country estate of hundreds or thousands of acres, whose lord typically had much but not unlimited civil and criminal jurisdiction over the occupants. Of course it was perfectly possible to hold a fraction of a manor as a fief. More important lords held several different manors as fiefs. Titled noblemen typically had over lordship over all the manors in their region, the manor holders being their vassals, as well as having their own manors. These noblemen would be in turn the vassals of higher nobles, and so on.
Even though the lowest lord of a manor exercised some governmental functions, that was not enough to make the typical feudal manor exactly a state and thus a vassal state, just as a municipal government does not usually count as a subordinate state.
And on the other hand, the very largest fiefs definitely counted as states of a kind and thus as vassal states. For example, the Kingdom of Thessalonica in the Latin Empire, or the Kingdom of Bohemia in the Holy Roman Empire. And in the colonial era, many large and powerful African and Asian states became sort of vassal states of various colonial powers.
I am not certain where to draw the line, or more logically a broad fuzzy zone, between ordinary vassal fiefs and vassal states. But it would be somewhere above the typical feudal manor and somewhere below vassal kingdoms and at least some vassal duchies.