A torque wrench is needed mostly for lightweight bike parts, especially at the high end.
There are several related causes: with lightweight parts the manufacturer has shaved off everything that is not absolutely essential.
There is no spare strength to allow for overtightening.
things are made very precisely now.
Rather than being able to take a 250kg gorilla, the lightweight bike can take a 120kg human.
And rather than be maintained by a gorilla, you need an actual bike mechanic.
This is partly to save weight, and partly to shave costs.
Why use 1kg of titanium where 500g will do?
Most people do things up until they feel the resistance change.
This slight yeild in steel is non-destructive and can be repeated thousands if not millions of times.
In aluminium you need to be more sensitive because the yeild limit is sharper, but with composites and lightweight metals like titanium and magnesium the change is so fast it might as well not exist.
The torque limits are likely to be tight and precise. to save weight more threads are cut into expensive parts.
Rather than having a semi-captive nut the body of the bike/fork/wheel is threaded.
Stripping it means replacing something expensive.
What this means is that the traditional steel bolt in a steel bike will work well with (say) 15-50Nm of force used to tighten it.
At 15Nm it's only just tight, at 50Nm
it's probably digging into the frame
and it's about to strip.
But a (hollow!) titanium bolt in a carbon/epoxy stem will have a torque range of 5.5-6.4Nm, and the failure mode at the high end will be the stem failing-
either it will strip the thread or the clamp will snap.
And the bolt is built to match - why put in a bolt that will take 10Nm of tightening when the stem will fail at 7Nm?
There are other design changes to match these - four bolt handlebar clams on stems are now commonplace where there used to be one bolt
and you just bent the clamp open then forced the (curved) handlebars through it.
You can't bend an aluminium clamp like that, it will fail.
And splined cranks require a lot more precision than tapered one, but that precision also allows them to be lighter as well as easier to service.
The days of stripped extraction threads in cranks are (hopefully) over.
bicycles.stackexchange.com/users/None/
Crank bolts.
They need to be really torqued, but you're always wary of over-torquing, so a torque wrench provides a good confidence factor.
bicycles.stackexchange.com/users/1584/Daniel R Hicks
I bought a torque wrench after writing off a the bottom part of a set of Fox 36's: Those four little bolts at the bottom hold the axle in place, and need to be tightened enough to stop it coming out, but not so much that you stress the clamp.
I'd over-tightened them and apparently the fork legs had tiny cracks.
The replacement legs have a metal shim to prevent over-tightening, and the design is now different - lesson learnt for me and for Fox.
The problem is that no-one buys a torque wrench when they're starting out and could really benefit from one.
bicycles.stackexchange.com/users/1534/Jonathan Sayce
I love my Torque Wrench.
It takes away the guesswork of being sure you are tight enough.
I picked up one from and off brand that was calibrated in inch/
lbs,
and it wasn't as expensive as the park version.
If you are no pro and working with carbon fiber, you need this tool.
bicycles.stackexchange.com/users/1627/Mario
I admit I never use one...
Still, for certain applications, specific tightening torque is specified.
Things like crank-retaining nuts, quill-stem retaining bolts...
Crank bolts on models with removable chainrings...
That sort of thing.
Some people have a good feel for "tight enough", but others may be ham-fisted enough to break or strip fasteners without some reference.
bicycles.stackexchange.com/users/1297/M. Werner
I am with yourself - no need (unless you have a very expensive pro roadbike).
However there are new 'carbon fibre' components that have changed since the good-old-days.
These can have things such as 'helicoil' inserts glued in to them.
In the olden days it was just chainring bolts that were tricky to get right without a torque wrench.
Nowadays you can do up almost every bolt to a torque level specified in the component's manual.
bicycles.stackexchange.com/users/1487/ʍǝɥʇɐɯ
I use a torque wrench because I like a little piece of mind, and have unfortunately ruined a few parts by over tightening.
I have a Wright Tool torque wrench which is perfect.
No complaints and no more broken components.
You can pick one up for a decent price off ebay.
bicycles.stackexchange.com/users/11406/Matt G
I was rather surprised to find a question about  choosing a torque wrench  on this site. I haven't cycled for years, but many years ago when I actually cycled I would do with a set of ordinary wrenches just fine.  What bicycle maintenance tasks would require a torque wrench and can't be done with an ordinary wrench?
Why would I need a torque wrench for a bicycle?
bicycles.stackexchange.com
<maintenance><tools><torque-wrench>
bicycles.stackexchange.com/questions/4120
bicycles.stackexchange.com/users/628/sharptooth
A torque wrench is needed mostly for lightweight bike parts, especially at the high end.  There are several reasons for this: with lightweight parts the manufacturer has shaved off everything that is not absolutely essential; there is no spare strength to allow for overtightening; things are made very precisely nowadays. This is partly to save weight, and partly to shave costs. Thus, any excess tightening can cause damage to the bike. On the other hand, you don't have to guess if you have tightened enough or not. The torque wrench avoids these problems. High-end parts are built to be as light as possible with a narrow range of tolerance that may not be clear from feeling the resistance. If this resistance is exceeded, carbon fibre and other materials can break, destroying the whole part. A torque wrench sets the correct torque, provides peace of mind and removes the guesswork.
High-end parts are often made as light as they can be, and therefore have less tolerance for overtightening. They are also often made with softer materials with a narrow range of acceptable tightness and more chance of breaking if this is exceeded, without this limit necessarily being clear from the resistance. Most people are not able to judge the correct resistance for a particular part, so a torque wrench takes away the guesswork.