The deep cycle capabilities of AGM batteries absolutely make the difference. With a standard cranking battery, the lead plates are very thin, which is what you want for lot of cranking amps. It allows for a whole lot of amps to be output in one whack. But those thin plates are also particularly bad for continuous draws, even low amp draws. A one amp draw can run a cranking battery down overnight to the point where it won't start, as will happen if you leave a dome light on, or certainly the headlights (55w bulb, in a 12V car, bulb draws 4.6Amps, times 2). Those thin plates don't allow for long term storage of usable amps like thicker plates of an AGM battery will, because of the high internal resistance of cranking batteries.
With a AGM battery (or any battery with thicker plates), you can have long low amps draws and still have plenty of usable amps for cranking. An AGM battery will not have the same cranking amps as a cranking battery of the same size, but that doesn't matter very often. You'll see a battery with 800 CCA, but the reality is that the cranking amps are determined by the starter. If you have a 400 amp starter, that's all the amps you'll need to start the vehicle (most starters are between 200 and 500 amps, but I can't speak to the starters of big trucks). But as important as amps are, it's the voltage to deliver the amps that's important, because it's the pressure of the voltage that pushes the amps around. As long as the battery is above 12.2 volts, it'll usually start the vehicle. If below that, it's iffy. But I've seen one as low as 11.8 volts start a vehicle.
The amount of power your starter motor pushes out is a function of the voltage and the current rushing through it (P = VI). That being said, the current flowing through the motor is a function of the resistance along the conducting path (Ohm's law, I = V/R). Your battery's voltage is always going to be around 12.5-volts, give or take, and the starter motor's resistance is just about 0.12 Ω (for a small to medium car or light truck). Thus, if we assume negligible resistance in the starter motor leads (painfully false assumption, but I'm trying to keep it simple), the current going through your motor is I = 12.5 V / (0.12 Ω + R_batt) where R_batt is the internal resistance of the battery. A more expensive battery (higher quality) generally means smaller internal resistance, a smaller internal resistance translates directly into higher current, and a higher current translates into a faster cranking speed.
AGM batteries have extremely low internal resistance. It's what enables them to recharge significantly faster, as very little voltage or current is lost to heat, and it's what enables them to deep cycle efficiently, and handle the low amps draws of forgotten headlights.
In modern cars, light trucks and vans, the role of the starter motor is to turn the engine over twice, so that the ECU can sync and then start pumping gas and igniting at the right time. A typical cranking speed for a passenger car would be around 350 rpm (in a Sprinter it's 400), so that means, in a modern car, your battery spends about half a second working each time you start it up, and then proceeds to do practically nothing for the rest of the trip.
During cranking, you need to power a plethora of other electrical components: engine sensors, the ECU, spark plugs, fuel injectors, fuel pumps... If these components don't get enough power because the starter motor is too much of a drain on power, then your car is simply never going to start, or it'll behave very erratically during cranking. Naturally, if you have less internal resistance at the battery, like with an AGM, you have less power losses, particularly when it comes to parasitic loads, and thus you have more available power.
To sum it up, my long-time mantra is, never skimp on tires or batteries.
If your vehicle is older, you'll have more problems starting it, so buy a really good battery.
In extreme cold, the engine will be harder to start, thus requiring more cranking time from the battery. Extreme cold also reduces the amount of cranking amps, so a low resistance battery like an AGM will perform better in colder temperatures. But a healthy regular battery will also perform quite well. You don't really need anymore more robust than an OEM battery, even in cold.
If your vehicle won't start, especially when it's cold, the cold has uncovered a problem, either with the battery that has gone bad or is about it, or some mechanical or electrical problem that needs to be addressed. It's not a bad idea to replace the wires going to the starter with a heavier gauge, which reduces resistance.
Generally speaking, more CA and CCA is better, but you should not need more than the OEM battery specs supplies. More than the OEM, is insurance. I have a direct replacement that has the same CA and CCA as the OEM battery in mine.
