What is difference between I2C and TWI?

I agree with everything @Wouter said... Namely, TWI is used by manufacturers (like Atmel) to refer to their I2C interface to avoid legal trouble with Philips since I2C is a registered trademark. Some manufacturers (like Atmel) have implemented proprietary features on top of I2C so that their TWI is an I2C superset.

To answer your follow-up question, yes. It is enough just to change the name. A trademark only protects the name. If it were patented, the situation would be different.

However, it is difficult to defend a patent on a modulation/encoding scheme since it is mostly just math. Often, copyright is used for this, but it is usually pretty easy to circumvent, since the copyright protects only the exact specific instantiation (implementation) of the protocol, not the protocol itself in most cases.

From i2c-bus.org:

TWI stands for Two Wire Interface and this bus is identical to I²C. The name TWI was introduced by Atmel and other companies to avoid conflicts with trademark issues related to I²C. A description of the capabilities of TWI interfaces can be found in the data sheets of corresponding devices. Expect TWI devices to be compatible to I²C devices except for some particularities like general broadcast or 10 bit addressing.

Also, for now, there is also no TWI high speed mode.

If you have an I²C analyser it will work with a TWI set-up as well without modifications.

TWI devices typically don't conform completely to the I2C specification.

Notably, clock stretching is usually not supported.