Why does my board start smoking when I put in a higher battery same voltage?
A few thoughts:


If your board was designed to work with 3.7V in the first place, then connecting the larger capacity battery should not be a problem since both batteries are 3,7V. Was your board designed for 3,7V?
As @Alexander already commented, check if you accidently switched the + and -
The 'JST' connector is rated upto 2Amp, so quite some current is flowing to make it smoke. Is there a fuse on the board? Is there a reverse protection diode on the board? Did you try to power up the board again with the smaller capacity battery?
Since specs on websites like Alibaba are not the most reliable, you could check the voltage coming out of your battery, and check if it's really (close to) 3,7V. I couldn't find the battery number you posted. There are lots of batteries called 'AWT', but then still no results for that number. (20AH is huge btw.)




Edit: Proof that you have the polarity reversed!

So I took a close look at your battery, and at the 1000C module and at the JST 2-PH connector.



This is a closeup of a generic JST 2-PH connector. Don't mind the colors of the wires as this is just a picture from the Internet. What's important is: You can clearly see that when the 'springs' that hold the contacts in the connector are on top, the '|' and two 'bumps' that hold the connector in the receptacle are on the bottom.



This is a closeup of the Adafruit 1000C module. The '|' and the bumps are on top, so the 'springs' must be on the bottom. With the '|' on top, the + is on the right. Put the '|' on the bottom, and thus the 'springs' on top, the + would be on the LEFT.



Now here's a closeup of the picture you posted in your question. The 'springs' are clearly on top, and you can see on the bottom there is one of the 'bumps', that are on the same side of the '|'. So the '|' is on the bottom. 'Springs' on top, + should be on the LEFT, but is on the RIGHT!

So your polarity is indeed reversed! You can carefully lift the 'springs' and take the contacts out. Then you can put them back in the right place. Be careful not to bend the 'springs' too far, as they are only plastic, and easly break off. Also make sure you don't accidentally short-circuit the battery when both contacts are out.

Hope this helps!

Update: I just saw they actually have a warning on their website for third party batteries with the polarity reversed!

How do you intend to charge the battery? Via the 1000C? Because that module can only charge with 1000mA max., so charging your battery (when completely discharged) will take more than 24 hours. I don't even know if you can charge a Li-Ion/Li-Po battery with less than 0.5C. But that's a completely different topic anyway...