Why does smartphone GPS find its position much faster than a GPS module?
There are several things which affect the time to first fix (TTFX). 


Getting the almanac and ephemeris. These two things are technically a little different from each other, but for our purposes we'll treat them as the same.  They are the locations of the satellites, and you need a to know where they are in order to work out your own position.  Each satellite transmits the whole lot roughly once every 12 minutes.  So from a completely cold start with a one-channel receiver and a decent signal, TTFX will be at least 12 minutes.  You can speed things up by:


Downloading from the internet instead - generally a good choice for phones. Downloading the almanac and ephemreris this way is known as MSB Assisted GPS.
Remembering the almanac from last time (it's good for many weeks) and only downloading the ephemeris.
Having more than one receiving channel in the device so you can listen to more than one satellite at once.  The transmissions are staggered to make this work, and with some care you can use the ephemeris without an almanac which saves a lot of time.  The vast majority of modules on the market these days have multiple channels, so it would be rare to find one which still needs 12 minutes.

Identifying satellites.  You need to listen to at least three satellites, preferably more, to get a good fix, but each receiver (known as correlators)  can only be tuned to one at a time.  If you know roughly where you are, what time it is, and have an almanac already, then you can guess which satellites you can see. Phones tend to know roughly where they are from recognising wifi or bluetooth signals, knowing which cell tower they are using, and other sources. They regularly get very accurate time updates too, so they can usually go straight for the correct satellite. Both phones and larger modules can also remember when and where they were last used, and use that to start from.
Number of correlators. Due to the very low signal-to-noise of GPS signals, you need a special bit of hardware to receive them.   Some receivers only have one, and need to rotate 'round the satellites.  Others have more, and can listen to more at once. So even if you already have the almanac/ephemeris and know roughly where you are, then more correlators will still help you fix quicker. You might think more is always better, but more does increase cost and power consumption.  Some phones and modules have more than others.
Signal and antennas. The correlators will do their job faster if you have a good signal-to-noise going into them. Very poor signals might not work at all. A good antenna design, amplifier, sky view, and good PCB layout can make all the difference.  Some modules may work OK out of the box, and much better with an antenna plugged in.
Number of usable satellites. There are actually two large constellations of satellites up there, GPS (run by the USA) and GLONASS (run by Russia).  There are also more under construction: Galileo (EU) and BeiDou-2 (China) and some with local coverage like India's NAVIC or BeiDou-1. A receiver which can work with satellites from more than one constellation has more satellites to chose from, and will get a quicker and more accurate fix. 
Quality of correlators.  New hardware designs are better than old ones, and will be able to pick out fragments of the GPS message in a noisy signal better.  Another trick phones can do is to capture fragments of signal and pass them over the internet to a server with a very good software correlator, and complete almanac/ephemeris to examine.  This is known as MSA Assisted GPS.
Some phones (and even a few modules) might also use some slightly sneaky tricks to avoid or hide a long TTFX. Since they are on all the time, they might briefly switch on the GPS without telling the user in order to keep the location and ephemeris roughly up to date.  Others might display a recent position while still waiting for a real fix - which looks like a good TTFX most of the time, but looks bad if it turns out the position is very wrong.


Point 1 above is the thing that makes the most difference, and is usually the key thing that is different between basic modules, more advanced modules, and phones.  The others usually make a smaller difference, but it can actually become a very complicated thing. If you want to read more, then "GPS time to first fix" is the term to search for.