Hmmm... With just the watch it's trickier, as it's easier to "fool" by changing your arm-swing. I'd still trust the watch more than the TM, without more data, though. I've run on a lot of treadmills, very few have been accurate.

I run on the TM about 50/50 during the winter. So, I got a footpod years ago. I've done a lot of careful comparison using the footpod, heart rate, and rpe to validate the indoor pace vs outdoor pace. My TM is off by 60s/mi (slower... In truth it varies with pace). Since I run inside so much, I like to be able to compare paces and heart rate between inside and outside.

Gyms are also tricky because they usually have crappy airflow... Ie, no fans. That skews HR and RPE upwards making slower paces feel harder.

If you are going to do this regularly, I'd get a footpod and take some data to figure out which is right. If it's just the occasional one-off day, I wouldn't sweat it, and would just trust your watch.