What happens when connecting same voltage, same polarity batteries in parallel?

what happens when batteries of the same voltage and polarity are connected in parallel?

One battery will always have a higher voltage than the other and the higher will attempt to charge the lower. Depending on the battery technology, the voltage disparity and the internal resistances, a high current can flow causing overheating and even an explosion. Don't do this.

You can combine batteries by using a resistor in line with each battery positive and connecting the open ends together and to the load. This will allow the resistances to limit the current in no load conditions and provide a voltage to a load, determined by Kirchoff's Laws. However you will lose voltage across the resistors and they will also dissipate power.

An alternate method is to use two diodes. Each anode is connected to the battery positive and the cathodes are commoned and connected to the load. Whichever battery has the higher voltage will supply the load, alternating as each discharges and drops below the voltage of the other. Another advantage of this method is that it protects your load from inversely connected batteries. However the disadvantage is that you drop some voltage and dissipate some power across the diodes. The voltage for silicon diodes will vary from 0.6V to 1V depending on the current. You can use Schottky diodes which will drop the voltage to around the 0.3V mark at low currents.

A more modern alternative is to control a FET transistor to behave like a diode with a very low voltage drop. Several manufacturers like Analog Devices and Texas Instruments make Ideal Diode Controllers.

Given your application you should look at a composite solution, often based on the ideal diode, by looking up Load Share Controllers, again from several manufacturers including those two above.

I'd agree with everything Aubrey says. However, my employer, who operates many hilltop radio sites with battery backup, does have up to 4 batteries in parallel for increased capacity (they are 12V 100AH deep cycle lead acids). Each battery has its own fuse (rated 50A IIRC) which protects against the large currents that can flow if batteries are mismatched. In practice though, I've never seen one of the fuses blow in normal use. But if I was designing the systems, I'd try to do a less "Brute Force" method of load sharing.

While I agree with what Aubrey and David say, I've been doing a bit of research into batteries for radio and RV (caravan) use. It seems that with the advent of Lithium Iron Phosphate (LIFePO) batteries, some of the dangers of paralleling batteries seem to be mitigated by the use of BMS (battery management system) within the batteries. Basically, the BMS protects the batteries from over charge and also from damage from over discharge.

So it is likely safe to connect LiFePO batteries protected by a BMS in parallel. RVers seem to do it all the time. I would only do it with new batteries of the same manufacturer and rating, otherwise you will likely end up with the stronger battery trying to charge the weaker. Even if safe, it's a waste of energy. Fuses on each battery seem like a good idea in any case.