A “newton” is the international unit of force. A human adult’s maximum biting force ranges from 520-1,178 newtons depending on factors such as age and gender. It requires less than 200 newtons to bite through a raw carrot. This is well within the normal limits of the average person. Biting through a finger requires so much force that attempts often lead to partial-amputation; a bite that doesn't completely sever the finger. There are cases of fingers being bitten entirely off, but such an act requires extraordinary force, far greater than 200 newtons of carrot-cutting power.
You'll still have to bite through skin, tendons, and some flesh.
Skin, unlike "meat", is very elastic. It's similar to trying to bite a rubber band instead of a carrot.
There are a lot of factors that contribute to mechanical failure, applied force is only one :)
[EDIT] According to this study done in 1956 (Jamming of fingers: an experimental study to determine force and deflection in participants and human cadaver specimens for development of a new bionic test device for validation of power-operated motor vehicle side door windows) cadavers bones were tested at maximum applied force of 1886 N for the index finger and 1833 N for the little finger. In 200 jam positions, 25 fractures were observed on radiographs; fractures occurred at an average force of 1485 N.
So if it takes 1485 newtons to cause fractures (obviously greater force is required to completely sever through the entire fingers) then it does not take the same force to bite through a finger as a carrot. [/EDIT]
Was wondering if you were waiting to mark this as the answer. I re-read your comment and realized you may have been waiting on a hard number for human bite force as well as the force required to sever the finger.
For human bite force: the Wikipedia article on Orders of Magnitude tells us that human bite force, measured at the molars is averaged at 720 N. As far as understanding what it would take to sever a finger you must understand how measurements of hardness are taken (and the different scales used, but I won't go into that.)
According to the Wikipedia article on Hardness: Hardness is a measure of how resistant solid matter is to various kinds of permanent shape change when a force is applied. Hardness is dependent on ductility, elastic stiffness, plasticity, strain, strength, toughness, viscoelasticity, and viscosity. There are three main types of hardness measurements: scratch, indentation, and rebound. Within each of these classes of measurement there are individual measurement scales.
Scratch hardness is the measure of how resistant a sample is to fracture or permanent plastic deformation due to friction from a sharp object.
Indentation hardness measures the resistance of a sample to material deformation due to a constant compression load from a sharp object.
Rebound hardness, also known as dynamic hardness, measures the height of the "bounce" of a diamond-tipped hammer dropped from a fixed height onto a material. This type of hardness is related to elasticity.
All that being said, the manner in which the finger is severed, be it stripping of the flesh from the bone vs. a clean cut vs. blunt force crushing/obliterating, makes answering the question of 'how much' force is required to 'sever' a finger from the body difficult. I hope this answers your question in enough detail to dispel any skepticism that the human finger can be severed as easily as a carrot by the human mouth.
(For purposes of this answer I have not researched specific data regarding macroscopic molecular bonding of the biological materials that make up the human finger. Perhaps a thesis is in order?)