This is obviously USA.
The employment taxes (social security and medicare) amount to 15.3% of your gross wages. When you're a W2 employee, your employer pays half of this and withholds the other half from your paycheck.
When you're independent (1099) you're expected to pay the whole amount yourself, from the money you earn. You need to file a quarterly estimated tax payment to do this. You'll also need to pay quarterly estimates for your federal and state tax (in the same payments) because nobody's withholding that money from your paycheck.
Employers also pay such things as federal and state unemployment insurance premiums and workers' compensation premiums. If you're independent you won't have those things: you won't be eligible for unemployment benefits. You're self-employed. If you do hazardous work you might consider getting your own hazard insurance.
Those things are why independent (1099) contractor hourly rates are typically higher than employee rates. You should definitely earn a bare minimum of 7.65% more per hour for the same work when you're a 1099 contractor -- that's you're share of social security / medicare. You should ask for 12% or so -- e.g. $36 per hour rather than $30, or $75 rather than $60.
The good news: You're a small business. You can file a Schedule C (sole proprietorship business) with your income tax return. On Sched C you can write off lots of costs: marketing, some equipment, licenses, telecommunications, some travel, etc. You can also give yourself a Simplified Employee Pension Plan (IRA-SEPP), into which you deposit tax-deferred money for retirement.
You can also write off the costs of a home office, but be careful: the tax authorities like to audit those claims. I've never thought it was worth the trouble.
If you're a 1099 contractor, it's a good idea to get in the habit of submitting regular invoices to your customer (your employer). Find out who actually pays those invoices in your customer organization. Ask that person to check over your first invoice, and make sure everything they need to expedite payment is on there. Find out how often they write payment checks (e.g. every other Thursday, or on the 11th and 24th of each month, or whatever), and how far in advance you should submit the invoice so it can get approved and paid.
All in all the 1099 route is an excellent way to go if you're disciplined enough to save the money you need to file your quarterly estimated tax payments.