What are the differences between "share options" and "warrants"? Are they actually the same thing?

A warrant is a type of option.

Warrants are "a type of security that entitles the holder to buy a proportionate amount of common stock or preferred stock at a specified price for a period of years. Warrants are usually issued together with a loan, a bond or preferred stock --and act as sweeteners, to enhance the marketability of the accompanying securities. They are also known as stock-purchase warrants and subscription warrants." (see source)

An share option is a right to buy or sell a stock at a pre-determined price. These aren't usually in conjunction with a loan (or when they are, they are usually called warrants).

I have seen warrants in small businesses, as some initial founders loan money to the company. In this case, the lenders / investors received warrants that allowed them to convert the loans to shares at their option, at a pre-determined price.

This allows the lender to defer the decision as to whether they want to end up with payment of the loan (with interest), or shares of the company.

I've seen share options on major companies as "calls" or "puts" listed on the major exchanges. Calls are the right to buy a stock at a pre-determined price; puts are the right to SELL a stock at a pre-determined price. Calls and Puts can be Bought or Sold, giving you alternate ways to "bet" on the performance of the company, with associated risks.

For instance, if you think the value of a stock is going to go up, you can buy the stock. Or you can buy a call (for a fraction of the stock price). Or you can "write a put" (i.e. sell a put option).

The difference in these three methods has to do with how much is at risk. In the first case, the stock would need to go to zero for you to lose all your money. In the second case (buy a call), the call option is typically a limited time offer, and if the stock doesn't move to the "strike price" (your agreed upon price at which you can buy the the stock, given the option), then you lose your money.

The riskiest of all is to Write a Put, in that you are agreeing to buy the stock from someone at a fixed price at a later date AT THEIR OPTION. If the stock price falls, then you can lose a lot, because you may have to buy it from them at the higher price, and then you'll be holding something that is worth less than what you paid.

Calls and puts are good way to "hedge" your positions, or "take some of the money off the table" if you have some gains that you are happy with. They can be valuable instruments to cut risk, or they can be used in very risky ways.

Warrants end up being very similar to buying call options, except the consideration is often a loan.