Yes, front-suspension-only bikes are commonly referred to as hardtails.
As far as the benefits go, there are a number of factors to consider. Any bike is a trade-off between these factors.
Intended use
For purely on-road use, suspension is largely unnecessary (although those who've sampled the delights of Sheffield's potholes might disagree
).
The rider can provide plenty of shock-absorbance through slightly-bent arms and legs without too much of a detrimental effect in terms of rider fatigue.
For serious off-road use over bumpy terrain, suspension allows the wheels to remain in contact with the ground more of the time, thus improving traction (transferring pedal power into motion, braking, steering). In addition, the suspension absorbs a lot of the forces that would otherwise be transmitted to the rider, thus increasing rider fatigue.
Weight
For a given bike (i.e. same frame materials and components), a fully-suspended bike will weigh more than a front-suspension bike, which in turn will weigh more than a fully rigid bike. Whether or not the extra weight is worth it is largely linked to other factors such as better traction and decreased rider fatigue.
Cost
Suspension components cost more than rigid frames/forks (for a given bike). In addition to this, full-suspension bikes tend to have higher R&D costs (compared to rigid bikes, where the traditional double-diamond frame has remained pretty much unchanged for over 100 years). Therefore the addition of suspension either pushes up the price of the complete bike, or the manufacturer will spend less on other components (e.g. wheels, gears) in order to meet the same price point.
There are a lot of other areas for debate (e.g. power transfer) but I reckon these are probably the primary ones to consider when deciding what sort of bike you want.
