Why is connecting two different voltage sources in parallel dangerous?
As a conceptual model, a voltage source maintains its specified voltage even if doing so requires the supply of infinite current. Obviously that is not a component you can buy.
Connecting two set to different voltage sources with non-resistive wiring breaks their definition, so is an unanswerable problem.
In the real world, many devices can be approximated by the Thevenin equivalent model, which consists of an ideal voltage source in series with a resistor representing the internal impedance and response of the non-ideal source to load (there is a dual, the Norton model, which has a shunt resistor across a current source).
If you interconnect Thevenin or Norton source models, you can use basic circuit analysis to calculate the results.
Components such as batteries may be imperfectly approximated by these models. With care they can give you some idea if you are likely to run down, (over)charge, or explode interconnected parts but for obvious reasons it's an area where you want to allow huge safety factors.
As a practical matter, the internal resistance of batteries can be very low.
This is a little deceptive because when you study Norton or Thevenin circuits, your classroom-examples are going out of their way to use reasonable theoretical resistances, so that the calculated results are within reasonable bounds that don't make the student ask if he's doing something wrong. The real world isn't like that.
If a 12-volt car battery sags to 8 volts when cranking 1000 amps, what is its internal resistance? Pretty darn low.
Say you go to the golf cart dealer and he lets you take four 6-volt junkers that won't make it 18 holes anymore, and you accidentally grab an 8-volt one. You decide to parallel them anyway, giving a 12 and 14 volt pack. Those batteries are pretty violently going to try to murder each other. (Or to be more precise, the 14 volt battery will try to "charge" i.e. overcharge the 12 volt battery at high current, limited only by their internal resistance.) You can do the math but the numbers will be ugly.
Of course, diodes change everything. If you must parallel batteries, a diode per series string will solve a whole bunch of problems. It will cause asymmetrical loading of the batteries, but that is certainly better than the alternative.
With lead batteries you would need to worry about discharging the taller pack to levels which fatigue the pack (a unique flaw of lead-acid chemistry is that dipping it below 70% full will prematurely age it, the deeper, the faster.) So for instance your 14 volt pack would deplete nearly to 0% capacity before the 12V pack would carry much load, and that would quickly age it to death. At that point you'd want a little bit of intelligence to at least cut off the lead-acid pack before a damaging level of discharge. Or better, have a fairly sophisticated buck converter which monitors battery levels and draws down each battery in proportion.
This paralleling could be worked to your advantage with dissimilar packs. E.g. A reliable, indestructible 10-cell NiFe pack has a slightly higher normal voltage than a 6-cell lead-acid pack, but much higher internal resistance. So the diodes would block the lead battery for routine service loads the NiFe can handle easily. But when an engine is being cranked over, the NiFe would sag out, and the lead-acid pack would shoulder much of the load - just the kind of momentary burst load that lead-acid is perfect for.