A better cooler doesn't reduce the amount of heat going into a case (aside from perhaps a blower-style cooler exhausting heat directly out the back). It's simply moving that heat away from the GPU and into the air more effectively, to keep the card operating at a lower temperature. The same amount of heat is being produced no matter what cooler is used.
So, a 120 watt card should always be putting out around 50% more heat than an 80 watt card under load, at least when the graphics hardware is being pushed to its limits. If the CPU is holding back performance of the faster card, the difference in power draw and heat output might be lower. Neither of these cards is all that power-hungry though, so I doubt the heat output would be too much of a concern in most systems. You would want to make sure that your power supply is capable of delivering enough power to the card alongside the rest of your components though, and that it has a PCIe power cable that all cards over 75 watts (and even many 1050 Tis) will require.
As for the 1050 Ti vs the 1060 3GB, I would go with neither at this point if you are looking to buy new hardware. The 1650 SUPER has been out for a year already, and can be over 20% faster than a 1060 3GB, and over 80% faster than a 1050 Ti, while drawing about 100 watts under load. Just make sure it's the "SUPER" version of the card though, as the non-SUPER 1650 only performs roughly in-between a 1050 Ti and a 1060 3GB...
https://pcpartpicker.com/products/video-card/#c=476&sort=price&page=1
As for how the two cards you mentioned compare, the 1060 3GB tends to be around 50% faster than the 1050 Ti. The VRAM is a bit more limited, so one might be more likely to have to lower texture settings a little in some demanding games, but in most current titles it should be a lot faster. Still, with the 1650 SUPER available for as little as $165 brand new (US pricing) with 4GB of VRAM, from a price-to-performance standpoint, a 1060 3GB would need to be around $135, and a 1050 Ti around $90 brand new to offer similar value relative to the performance they provide. And for used cards, it might not be worth paying even that, as they will be lacking warranty coverage and may have been subject to a few years of use under unknown conditions, potentially affecting how long they last.