What is the difference between sin and karma in Buddhism?

Sin is a transgression against divine law (Wikipedia). Bad kamma is a transgression against natural law (dhamma niyama). For example, a word search for 'transgression' in some Buddhist scriptures brings 134 results.

Sin & bad kamma both include murder, theft, sexual misconduct, false speech, etc. They are, in essence, the same. For example, whilst being a translation, a word search for 'sin' in some Buddhist scriptures brings 251 results.

In Buddhism, Christianity & Islam, engaging in sin or bad kamma leads to 'hell'. For example, a word search for 'hell' in some Buddhist scriptures brings 609 results. A word search for 'hell' in the Bible brings 15 results. A word search for 'hell' in the Quran brings 110 results.

Oftentimes, bad karma will be from "doing something bad," but it can also be just from thinking something bad.

The difference between "sin" in Christianity and "deeds that reap bad karma" in Buddhism is that a sin, in Christianity and Islam etc., is a sin "against God." It is a transgression of a commandment from God. In Buddhism, there is no notion of a "God of Abraham," so you can't "sin against him."

Both religions believe in "good and bad deeds," but only one of them believes in "sins."

Christianity says sin is when someone does something bad.

That sounds like a circular definition. At the risk of being off-topic on this site, here is some detail.

Christian doctrine includes the concept of "sin" -- these include, for example:

Lust, Gluttony, Greed, Sloth, Wrath, Envy, Pride

And there are virtues -- perhaps (depending on the list) corresponding virtues -- for example:

In Christian tradition, the seven heavenly virtues combine the four cardinal virtues of prudence, justice, temperance, and fortitude with the three theological virtues of faith, hope, and charity.

The seven capital virtues, also known as contrary or remedial virtues, are those which stand opposed to the seven deadly sins. They are often enumerated as chastity, temperance, charity, diligence, kindness, patience and humility.

In my opinion the "sins" more-or-less correspond to Buddhist Kleshas (or kilesa):

Kleshas (Sanskrit: क्लेश, romanized: kleśa; Pali: किलेस kilesa; Standard Tibetan: ཉོན་མོངས། nyon mongs), in Buddhism, are mental states that cloud the mind and manifest in unwholesome actions. Kleshas include states of mind such as anxiety, fear, anger, jealousy, desire, depression, etc. Contemporary translators use a variety of English words to translate the term kleshas, such as: afflictions, defilements, destructive emotions, disturbing emotions, negative emotions, mind poisons, neurosis etc.

The theology may be different, but the lists or identities are similar, for example:

greed, hate, delusion, conceit, wrong views, doubt, torpor, restlessness, shamelessness, recklessness

... or ...

Attachment, Anger, Ignorance, Pride/Conceit, Doubt, Wrong view/False view/Opinionatedness

There are Buddhist lists of virtues too.

Christianity theology doctrine talks about sins being a deliberate distancing of oneself from God.

Buddhism is different -- there is no "God" in the same way, and no "God's law". But there is the Dhamma, the teaching, which might be called natural law -- I think it's meant to be self-evident, or evident to the wise person who looks into it:

The Dhamma is well declared by the Bhagavā: visible here and now, immediate, inviting to come and see, effective, to be individually ascertained by the wise.

In my opinion a mental state like "anger" is unfortunate -- it's unpleasant, and with unpleasant consequences all round -- based on, like, personal experience.

Christianity identifies "pride" as the original sin:

Pride (superbia) is considered the original and most serious of the seven deadly sins on almost every list. It is the most angelical or demonic out of them.[40] It is also thought to be the source of the other capital sins, known as hubris (from Ancient Greek ὕβρις) or futility. It is identified as dangerously corrupt selfishness, the putting of one's own desires, urges, wants, and whims before the welfare of other people.

In even more destructive cases, it is irrationally believing that one is essentially and necessarily better, superior, or more important than others, failing to acknowledge the accomplishments of others and excessive admiration of the personal image or self (especially forgetting one's own lack of divinity and refusing to acknowledge one's own limits, faults, or wrongs as a human being).

The Buddhist analog to this seems to be "conceit" (identified as a cause of people arguing), but the Buddhist analogs of the "original sins" are probably the three poisons and especially "ignorance".

In Buddhism, people are reincarnated for karmic reasons

One of the explanations that I've read on this site is that people are reborn from moment to moment, and that the moment of death is no different (i.e. there's rebirth after that moment too).

There's some doctrine -- which pre-dates Buddhism, so far as I know -- that "good karma" leads to a favourable rebirth ... perhaps as a high-caste human, or in a heaven ... and bad karma leads to rebirth in a lower/suffering realm, as a hungry ghost or animal or in hell.

To end rebirth I think you have to end karma -- see also Four stages of awakening

Another difference between Buddhism and Christianity is I guess that, from a Buddhist perspective, a Christian wants to be reborn in heaven and expects to then stay there for eternity. Conversely Buddhism teaches that all conditioned states are impermanent, and that includes a stay in heaven ... that eventually the kamma which led to your rebirth in heaven runs out, and then you're reborn somewhere else. This more-or-less unending cycle is called Samsara, and a Buddhist ideal is to end rebirth. Many people don't subscribe to that ideal, and consider it either undesirable or unattainable, and might aim for some rebirth in heaven (or with loved ones) instead.

What is the difference then between sin and karma (or what ever that causes bad karma)?

Simplistically, using examples:

• Murdering your parents and therefore being reborn in hell seems to me an example of sin and of kamma
• Giving generously to the poor, with the intention of being reborn in heaven, is NOT a "sin" but IS an example of "kamma".

This is a difficult question to answer because there's so much Buddhist doctrine. I haven't even mentioned the Pratītyasamutpāda for example, which may be key to the doctrine of how "birth" comes to be.

Keep in mind, though, the thicket of wrong views. Doctrine about rebirth and so on is attractive or perhaps scandalous, depending on your upbringing, however,

[...] Bound by a fetter of views, the uninstructed run-of-the-mill person is not freed from birth, aging, & death, from sorrow, lamentation, pain, distress, & despair. He is not freed, I tell you, from suffering & stress.

The well-instructed disciple of the noble ones... discerns what ideas are fit for attention, and what ideas are unfit for attention. [...]