Generally, you need a smart hub when some of your devices don't use Wi-Fi and can't communicate with your home network directly. Quite a lot of devices, especially sensors and actuators (like door locks and motors), use other protocols such as ZigBee, Z-Wave, Bluetooth and Thread, so they can't 'speak' directly with your Wi-Fi router. The hub acts as a translator, being able to communicate with both Wi-Fi networks and other protocols (almost always ZigBee and Z-Wave, but sometimes others too).
What's the difference between Wi-Fi and these other protocols?
Wi-Fi works well for applications where you're transmitting a lot of data, throughout your house. For example, if you're streaming a video across your house, you'll have no problems at all using Wi-Fi. But, as explained by Time, Wi-Fi isn't perfect for every use case:
"Wi-Fi is a whole-home network,” says Chris Coley, principle engineer and architect with Logitech. Primarily used for media streaming, browsing the web, and other data-heavy activities, it’s a high-bandwidth network that’s power-intensive — just watch how fast your laptop battery dies when you’re watching a video on Netflix.
Many smart home products eschew Wi-Fi-connectivity because it would require their devices to have a dedicated power source or a long-lasting battery.
It'd be very costly if every networked device in your home used Wi-Fi, and your shiny new LED bulbs might use even more power than old, incandescent light bulbs, so devices like this turn to other protocols, such as ZigBee or Z-Wave. Philips Hue bulbs, for example, use ZigBee, which is why the bridge is needed to 'translate' back to Wi-Fi so all the other devices can interface with it.
You can see how ZigBee and Bluetooth compare on page 51 of A Comparative Study of Wireless Protocols: Bluetooth, UWB, ZigBee, and Wi-Fi; in Figure 6, you can see that Wi-Fi consumes far less power per megabit. So, why don't we use Wi-Fi for everything? If you look at Figure 5, you can see that each Wi-Fi transmission uses far more power than ZigBee (Wi-Fi uses about 700mW, compared to ZigBee's power usage of less than 100mW).
For your light bulb, or thermostat, Wi-Fi doesn't make a lot of sense. If your device is just sending a few bytes worth of data (e.g. just a number), why waste loads of power on each transmission?
In short: Wi-Fi makes sense when you're sending large amounts of data across long distances. Your devices probably don't need to do that, so other protocols make far more sense.
What else do the hubs do?
Some hubs, like SmartThings, can run pre-programmed routines ('automations') and allow you to customise their behaviour. Often, devices provide IDEs or editors so that you can customise the rules.
You might be able to integrate with IFTTT with some hubs (particularly Wink and SmartThings), which might be the simplest way of linking services together if you're not able to program using your hubs.
Do I really need a hub?
This really depends on your personal setup. If all of your devices use Wi-Fi, then no, it's probably not necessary. One FAQ on Reddit has a concise explanation:
A close parallel can be drawn between a hub and a standard Wi-Fi router. In simple terms, both are boxes with radios and a little bit of smartness. Eventually, hubs and Wi-Fi routers may merge into one product (indeed, the Almond line of routers from Securify already do this), but currently most households will need to buy a hub because no other device in the house has the requisite radios to talk to the various switches and sensors. In addition, the hub must remain in the house so that automation can continue even when no person is there. This allows for presence simulation in “vacation mode” or pre-heating or pre-cooling the house before arrival.
In effect: lots of hubs don't just 'translate', they can also perform some processing and automate things that the manufacturer didn't provide by default. If you want the flexibility of customising how your devices work and interact, a hub is a great idea. If you're just planning to have a voice assistant (like a Google Home or Amazon Echo) and want to fetch the weather, there's probably no need just yet.
To figure out which hub you should pick, consider the following points:
Which protocols do your devices use?
You should be able to search for hubs which are compatible with a specific protocol. For example, when I searched for "Z-Wave hubs", I found this list of products, so you can get some additional guidance.
Do your devices list specific integrations with a hub?
For example, the ecobee3 specifically lists SmartThings and Wink as compatible; you should factor this into your decision.
Does the hub you're looking at allow you to automate things simply?
Some hubs may use a programming language you're unfamiliar with. It's probably best to avoid them, otherwise you're stuck with a device you can't control!
So, if that was all too much to read:
Bridges generally just 'translate' from one protocol to another, and tend to only work with one device or manufacturer (e.g. the Hue bridge).
Hubs can both translate and run automation routines themselves, allowing your devices to interact in a more complex way.
For anything more complex than a trivial setup, you probably need a hub. Make sure you research to find out which one suits you!