General question:
What will I get from an expensive flash unit, that I can't get from a cheap one?
Specific example:
The $250 Canon Speedlite 430EX II vs. the $50 Neewer TT560. To my naive eye, the flashes look pretty similar. What is better about the first one that justifies a 5x higher price?
With a more expensive speedlight/flash you typically gain:
None of this is to say that an inexpensive manual flash isn't a great option. They really are, especially if you are trying to learn and really understand light. If you want a fully automatic experience or have significant needs around professional reliability and features - then the OEM/name brand units are better. If you are just starting out and are interested in learning how to light, then the off brand cheap guys are wonderful and I highly recommend even the one you linked to.
When you're selecting between the OEM (original equipment manufacturer; e.g., Canon or Nikon) and a 3rd-party flash (Metz, Sigma, Nissin, Youngnuo, Neewer), the main thing that's different is that the OEM designed and knows the internals of the flash/camera communication protocol, while 3rd parties generally reverse engineer the electronic signaling protocol between the camera and the flash.
That means that typically a 3rd party flash won't have the same forwards and backwards compatibility of an OEM flash. Which is why some of the higher-end 3rd party manufacturers include the capability to upgrade the firmware on their newer units. When Canon decides to add some whiz-bang new features (say, groups D and E, ID codes, Groups mode, and RT communication as they did with the introduction of the 600EX-RT and ST-E3-RT), they'll make sure their own older flashes are still compatible, but they don't really care about 3rd party units not working. A 3rd party flash will probably be compatible with the system as it stands today—they may not be so good with how the system was ten years ago, or will be ten years from now. A digital-era Canon or Nikon OEM speedlight is still compatible with their film SLR bodies.
Case in point. I have a Yongnuo YN-568EX that I love on my dSLRs. But if I try to put it on my Powershot G9's hotshoe, it won't even sync (fire). My Canon 580EXII works just fine on both types of cameras, and can even have its power adjusted via the menu of the G9.
3rd parties also often have less name recognition. And the super-cheapies (e.g., Godox, Yongnuo, Neewer) typically don't have the same consistency as the OEMs in copy build quality, or components. And their warranty and service isn't at the same level, often relying on the retailer you purchased from to offer support. This is why they're cheaper. And this is also why OEM gear is going to have better resale value.
Whether you need that dependability is up to you. If you're a hobbyist weekend shooter, you may not see a thousand flash bursts for months. If you're a working pro who does event or portrait shooting, you may see that in a single day. There are reasons the pros don't go Yongnuo for their flashes (or triggers). But not everyone needs PocketWizards any more than they need Profotos.
The main differences between the two flash models you've chosen are that the 430EX II is a mid-grade eTTL-capable OEM flash, while the Neewer is a lower-end manual-only 3rd party flash (it's actually a rebranded Godox TT560). There are reasons to go with either, but for a first or only flash, the 430EX II is probably the better choice for the following reasons:
TTL—This is an automated way for the camera to set the flash's power level. The camera tells the flash to send out a preburst flash of a known brightness level, meters it through-the-lens (TTL), and then adjusts the flash's output based upon the reading (and within the power limits of the unit). Think of it as the flash analog to having Av mode on the camera body, as well as M. Nearly all TTL flashes can do M; the reverse is not true. TTL becomes useful when you do run'n'gun event shooting, where you don't have time to chimp adjust and reshoot until you get the flash power right. The Neewer is manual-only. You'll have to dial in the flash power every time.
Manufacturer warranty, service, pricing, availability and resale value are better.
Canon wireless slave mode built in. If you're shooting a T3i or later dRebel, a 60D or later XXD model, or a 7D or later, the pop-up flash of your camera has a built-in master that can command the 430EXII via its built-in slave sensor. The TT560 doesn't have that sensor.
Flash-camera communication. The 430EX II can do it, a manual only flash can't. For Canon cameras, this means 2nd curtain sync, high-speed sync (using your shutter speed over the X-sync speed of your camera body (typically 1/200s or 1/250s), auto zooming based upon the lens focal length, using distance information from the lens, being set from the camera menu, etc. are all things the 430EX II can do that the Neewer can't.
That's not to say all 3rd party options are manual-only cheapies. 3rd party flashes from Godox (at the lower end) or Profoto (at the higher end) are actually preferred over OEM offerings by many shooters. There is a spread in terms of the reputation and features from 3rd party flash manufacturers.
You can purchase a 3rd party Godox flash that has a feature set similar to a Canon OEM speedlight: TTL, HSS, built-in radio remote control, etc. with additional features Canon might not offer, like PC sync ports and "dumb" optical slave modes (that don't require a master unit, just a flash burst to pop off the remote flash). But they're still going to have the 3rd party weaknesses outlined up above, and the more feature-laden units are priced higher, so they're not so cheap any more. A Profoto A10 has a great reputation for reliability and function but costs the same eye-watering $1k pricetag as the top-of-the-line Canon EL-1.
So what you're juggling is low-cost vs. features vs. reliability. You can have only two things on that list. :) And only you know which two are most important to you.
It may be worth it to you to get a $60 all-manual flash if you just want to try shooting Strobist-style to see if you like it. It's probably not worth it if you're a pro with clients breathing down your neck and your flash gear absolutely has to work flawlessly every time so you end up buying four copies just to make sure you've got backups.
Usually for each Canon/Nikon brand flash you can find 3rd party alternative which is 2x-5x cheaper and provides a similar functionality. The difference is mostly in build quality, and servicing options. For example: cheaper plastic, plastic vs metal hot shoe, light indicators vs lcd screen, battery door design, etc. If it brakes you might have to ship it to China.
One real difference is wireless eTTL/iTTL in the top Canon/Nikon flashes. That's where 3rd party really lags behind. But we are talking about $500-$600 flashes here.
For the seasoned pro, there are (possibly) faster recycling times, more stable flash duration and longer battery life. If you have to worry about them, you don't need a flash advice :-)
Don't go for the cheapest options, do your research. There are lots of complaints about flash being DOA or dying in a month. Check flickr discussions for example. Go for a reliable 3rd party brand that many people use. For the same price you'll get 2-3 flashes (i.e. 2x-3x more light and flexibility).
