This is a fairly complex topic. I prefer shares to options myself, but it's very situation dependent. Sometimes options are the only way you can avoid paying a huge amount of tax on shares that you can't sell yet. Also, figure out whether you're getting ISOs or NQSOs, as they're taxed differently.
The high-level summary is: options provide you with the *gains* from a set strike price, and you're taxed on those gains (usually at the ordinary income tax rate) when you exercise the options. Exercising gives you shares, and you can then choose to hold on to those or sell them immediately. If the shares rise in value, you pay capital gains taxes on the rise, just like you would with regular stock.
If the stock drops below the strike price, your options are worthless!
Shares, on the other hand, are pieces of the whole company, not just the part that grows after you join. They therefore have value at grant, whereas the initial value of an options grant is usually $0. They're taxed either at grant (if you make an 83(b) election), or at vest (if you don't). Shares can drop below the grant price and still have value, albeit probably less than you were originally expecting. Usually (but not necessarily) they carry voting rights, whereas options don't.
In a small startup, it's almost always most favorable to take the shares and make an 83(b) election within 30 days of grant, since the value of the startup is very small and you'll be taxed at the much more favorable long-term capital gains rate if you sell more than a year later.
In a company that's already large, coming into a large amount of stock is a windfall - unfortunately, you probably can't afford the taxes on your unvested stock since it isn't liquid. In this situation, you can either have the stock taxed at vest (i.e. no 83(b) election) and run the risk of a paying more taxes later, or take the options and lose out on capitalizing on the existing size of the company.
* I'm not a tax professional and you should seek one if you're looking for tax advice.