I have both the 45-250 and the 100-300. They're quite different lenses.
The 45-250 is quite light and easy to add to the gadget bag when walking around. On the other hand, I don't think it's a particularly great lens at any of its zoom range. I have manual focus lenses at 45, 85, 100, 135, 200, and a couple more between those, and I don't think the 45-250 comes close to any of them in terms of image quality. On the other hand it's very convenient, so I still take it with me at times.
The 100-300 is a total surprise. At 135 or 200, I've gotten really amazing pictures from it from the start, and having 300mm f/5.6 (600mm equivalent) in your hands, with OIS and autofocus is a great experience. On the other hand, it's significantly bigger, significantly heavier, and unless I'm pretty confident I'll be wanting a telephoto, I don't just "throw it in the bag".
I originally bought the 100-300 for a trip to Rio over Christmas, but chickened out and took the 45-200 instead. (I had barely even tried out the new one when I left.) When I got back, I started using the 100-300 seriously, and totally regretted my decision.