Whether you're a 1099 contractor (what you initially describe) or a W2 employee is as much a matter of fact as a matter of what's easier or cheaper.
At least in theory, a contractor should be someone who is largely in control of things like schedules, you often provide your own tools, etc.; you are performing a specific service for the company, but how you perform that service is up to you (within obvious limitations, such as budget and project deadlines). You may work for many companies at once.
An employee is someone who is expected in the office at certain times, is expected to follow corporate procedures, may be told how to do their job in addition to what the end results should be, and uses tools provided by the company. You probably only work for one company.
While in some cases you may have the opportunity to be either one, much of the time only one of those descriptions fits. Think of a plumber: if you are called to the work site at need, and tell them when you can arrive, if you bring your own plumbing tools, and if you can reject work when you're too busy, then you're a contractor. If you are working at the company 9-5, are told when to be where, and use the company's tools, then you're an employee.
They don't necessarily have to offer you all of the same perks as a full employee - that's up to the company to determine how you qualify for their benefits, but especially post-ACA you likely would have to be offered health insurance at a minimum unless it's a very small company. Some benefits (like Workers Comp eligibility) accrue automatically as part of being an employee.
As far as the company goes, there are significant additional taxes required for a W-2 employee, both for the employee and the company. Medicare, Social Security, Unemployment, and Worker's Comp insurance are all needed for a W-2 employee from the company; for a 1099, you pay Medicare/Social Security yourself (and pay a higher portion, as the employer usually pays a part of that) under Self Employment taxes, if you made more than $400. Companies would certainly prefer you to take 1099 status, to the point that billions of dollars per year in penalties and back taxes end up owing due to misclassification of workers as 1099 when they shouldn't be.
See the IRS's tip sheet on determining employment status for more information.