Short answer: There is no difference.
Long answer: Come, ride with me through the veins of history. I'll show you a god who falls asleep on the job. (And how can we win, when fools can be kings? Don't waste your time or time will waste you.)
The word 'company' (as in a corporation) is derived from the word 'company' (as in a group of people, like the Company of the Ring). The word 'corporation' reflects the fact that the organisation has an independent existence for legal purposes, that is, it metaphorically has a 'body' ('corpus') or has been 'incorporated'. (If you have ever wondered why corporate culture is so boring then consider the common etymology of the words 'corporation' and 'corpse'.)
Before there were incorporated companies, there were 'joint stock companies'. The term 'joint stock' referred to the fact that members of the company (analogous to what we would today call shareholders) jointly held the 'stock' (as in 'inventory') of the company. This use of the term 'stock' to describe a type of company makes no sense in today's language, but originally the term was used to distinguish a 'joint stock company' from a guild (also known as a company) where the members each maintained their own inventory. For example, in the Worshipful Company of Stationers and Newspaper Makers (the printers' guild in London, which managed censorship on behalf of the government in exchange for the right to suppress piracy), each printer had their own stock of paper and finished products; in contrast the Muscovy Company had a single stock of wool that it took over to Russia and a single stock of furs that it brought back to sell in London (the 'joint stock').
(The early joint stock companies were of course technically partnerships, which is why, in the Income Tax Assessment Act 1997, the definition of 'partnership' excludes a company and the definition of 'company' excludes a partnership, forming a neat circle which everybody ignores in practice because they want money. Incidentally, as you would imagine, a limited partnership, which may be incorporated, is a partnership rather than a company for tax purposes, unless it is a 'corporate limited partnership' in which case it is a company. But that is tax law, you must never go there Simba.)
So anyway, back in the day, if you had put 10% of the capital into a joint stock company then you could be said to have a right to 10% of the 'stock' of the company. Over time, as the trade in interests in companies evolved, the term 'stock' came to refer to a member's rights in respect to the company, e.g. 'I own stock in the XYZ Company.' Since what you hold is a share in the stock of the company, it was said that the member held a 'share'.
Subject to company law and the company's constitution, etc, a company could split or merge shares. So, if you had 100 shares in the company, the company might split each share into 10 shares, giving you 1,000 shares.
Back before we had electronic computers and the Internet, keeping track of shares was a lot of work. If a person had 1,000 shares then you might have to move around 1,000 printed share certificates. Therefore the practice developed of converting a person's shares in a company into stock. Stock representing 1,000 shares could be documented as a single item instead of 1,000 items. (Of course, you could still sell one 1,000th of that stock if you wanted to.)
Of course a shareholder could never convert their shares into stock. That would make no sense. Shares (or stock for that matter) do not exist outside of the records of the company. Therefore the company, or more precisely the directors on behalf of the company, must be the ones to convert shares into stock or vice versa.
The terminology that developed, and which persists to this day outside of strict legal contexts, is:
- a 'share' is a single unit of ownership in the company;
- 'stock' is the whole of a person's ownership in the company.
In Morrice v Aylmer (1875) LR 7 HL 717, the House of Lords confirmed that a bequest (in a will) of the deceased's 'shares' would imply a bequest of their 'stock'. So, even in a legal context, there's not much of a distinction.
The one distinction of note is this: you cannot have partly-paid 'stock'. If shares are only partly-paid, then you have to keep track of them individually. You can't have the company converting a bunch of shares to stock, not distinguishing between which are paid up and by how much, and then converting them back into shares, with who-knows-what consequences (probably none to be frank, but we're talking about 19th-century information processing systems here). Therefore there was the rule that you can only convert shares to stock if they are fully paid-up.
In Australia, in the 1995 draft of the Second Corporate Law Simplification Bill (written by the Attorney General's Department's 'Simplification Task Force') it was proposed to abolish 'stock' and force all Australian companies to convert stock into shares. This proposal became law in the Company Law Review Act 1998, and today is reflected in section 254F of the Corporations Act 2001, which says that a company cannot issue stock or convert shares into stock.