What's the difference between monks and priests?

So far as I know (I may be wrong), "priest" is only used of Japan.

I've read three different theories for that.

1. One is that the term is for Japanese monks who marry (or who may marry). Usually part of the definition of a Buddhist "monk" (in any country) is that they keep some version of the Vinaya (which includes celibacy). I think that Japan is exceptional, for historical reasons. Wikipedia's Buddhist monasticism -- EastAsia currently says (see the original for hyperlinks and references),

   In East Asia, monastics live in greater isolation from the lay population than is observed in most Theravada countries. Because of local conditions of geography and climate, as well as local attitudes towards begging, monks generally do not make begging rounds in China, Korea, Vietnam, and many parts of Japan. Instead, monasteries receive donations of bulk food (such as rice) and funds for the purchase of food that is then stored and prepared at the monastery. Many monks and nuns are vegetarians and, after Baizhang Huaihai, many monks farm food to eat; some work or sell. Most eat after noon. The management of the kitchen and monastery properties may be the purview of a specially designated layman or a monk who has been given a special role by the abbot of the monastery. Monks chant many mantras in regular living.

   Monastics in Japan are particularly exceptional in the Buddhist tradition because the monks and nuns can marry after receiving their higher ordination. This idea is said to be introduced by Saichō, the founder of the Tendai school, who preferred ordaining monks under the Bodhisattva vows rather than the traditional Vinaya. There had long been many instances of Jōdo Shinshū priests and priestesses marrying, influenced by the sect's founder Shinran, but it was not predominant until a government Nikujiku Saitai Law (肉食妻帯) was passed during the Meiji Restoration that monks or priests of any Buddhist sect are free to seek wives. This practice influenced Korea and Taiwan. A nun in Taiwan gave birth. Some Korean monks live with wives in their monasteries.

2. The second is that "priest" is used for senior monks, who have graduated and are no longer students. Note the quote above, which says that "monks and nuns can marry after receiving their higher ordination".

3. Thirdly, some perform the role of priests in other religions (i.e. Shinto). Wikipedia's Tendai and Shinto says,

   Tendai doctrine allowed Japanese Buddhists to reconcile Buddhist teachings with the native religion of Japan, Shinto, and with traditional Japanese aesthetics. In the case of Shinto, the difficulty is the reconciliation of the pantheon of Japanese gods, as well as with the myriad spirits associated with places, shrines or objects, with the Buddhist doctrine that one should not concern oneself with any religious practice save the pursuit of enlightenment. However, priests of the Tendai sect argued that Kami are simply representations of the truth of universal buddhahood that descend into the world to help mankind.

In English (Christianity) the difference between priest and monk is that a monk lives in a monastery or priory (with other monks); while a priest will, more likely, live alone (in parish) to serve the lay community.

Householder Thomas Schulte, interested,

if looking into the texts this, Monks and Priests, is often mentioned as "Brahmans and contemplatives (Samana)", both regarded as people of respect in good societies.

The main different is in regard of livelihood. A monk, contemplative as given up all kinds of trades and lives from alms from the land. A Priest is one who still trades, involves in association with worlds societies and often has not given up harming totally, sense-pleasures and holding on home, land, house.

Most monks outside the Savaka Sangha and old tradition, even if often called monks (Bhikkhus) are actually "just" regarded as householders often not sticking to fundamental marks of one who has left home. Even many monks are actually "just" Brahmans and it's not easy that one would met a Samana, the fourth heavenly messenger after encountering the truth of suffering and can so imagine The Fruits of the Contemplative Life.

(Note that this is not given for trade, exchange, stacks, entertainment and akusala deeds, but as a share of merits and to continue such for release)