Moopla, I'll give it a try. This does simplify things a little so people please don't shoot me if it misses or simplifies things....
Basically a battery likes to be charged at around 13.8 - 14.5v, and with several stages like a 4 stage charger does for example.
However a Solar panel runs at between 17 and 22V for a so called 12V panel, and between 35 and 45v for a so called 24v panel. So you can't just connect a solar panel to a battery or it will fry it with over voltage. A charger or controller regulates the voltage from the higher solar panel down to the battery voltages.
The most common type of regulator is a PWM Pulse Width Modulation which quickly turns on and off to drop the average voltage down to say 13.8V. So if you have panels like yours, 175W and 40v at 5A, the regulator will "regulate" the voltage to say 13.8V at those 5A. The exact charging voltage will depend on the state of the battery charge, and the charging stage, 1- 4. These are only about 70% efficient on a 12V panel with the losses going to heat. However with a 24V panel the losses will be about 60%, and only about 40% or about 70W actually gets to the battery.
On the other hand, a MPPT regulator has 3 functions.
1)It will regulate the voltage and charge with say 4 stages like a PWM or a normal battery charger.
2)It looks for the sweet spot, or best performance, of your solar panel under current light and temperature condition. This is Maximum Power Point Tracking - MPPT. You will notice that the maximum power voltage of your panel is 36.2V and the maximum power current is 4.85A. Note 36.2 x 4.85 = 175.5Watts, that is the spec of that panel and it's best performance numbers. Most MPPT chargers will look for the sweet spot every few minutes or as light conditions change. Note the tracking refers to it looking for the best voltage and current, and does not refer to tracking the sun across the sky which is very different.
3)Here is the bit that helps you in this situation. Unlike a PWM regulator which regulates the voltage and can deliver the maximum current of the solar panel ( in your case about 5A), a MPPT charger internally converts the Solar DC signal to AC then back to DC. The reason for this is that it preserves the POWER of the input while regulating. So again in your case it would output 175 watts. At 13.8V that means it will output 12.5A ( 13.8V x 12.5A = 175.5W) in reality it would be about 12A due to inefficiencies but you get the idea. This is very similar to how a 12V to 240 v inverter works. A 480W inverter needs to draw 40A at 12V to supply 2A at 240V. ( 20 x 12 = 480W and 2 x 240 = 480W)
So the main difference is that while the controller you have can accept 24V panels, it will lose more than half the power to heat. Whereas a MPPT charger will be about 95% efficient in power transfer.
Which to buy?
There are lots out there and they ARE more expensive than PWM.
There are a few out there, especially on Ebay which are not true MPPT too and don't do the voltage and current step up / down. 'Wellsee' is a classic for that.
GSL is a good Australian brand and make a 12A one for $125 on ebay http://www.ebay.com.au/itm/GSL-MPPT-SOLAR-REGULATOR-PANEL-CHARGE-12A-12-24v-SYSTEM-/370493269008?pt=AU_Solar&hash=item56431ea410
Tracer make good efficient ones at low costhttp://www.ebay.com.au/itm/MPPT-10A-Solar-charge-controller-100V-lcd-meter-12V-24V-auto-Tracer1210-/250955548812?pt=AU_Solar&hash=item3a6e1dc88c
Perhaps others could suggest good brands at a reasonable prices too.
Hope that helps.