For most the part you should be ok with doing it either way. But, this depends on how fast of a connection you are going to be using to both the internet and the external harddrive itself.
First, when you download make sure you are wired in and not on wifi, you'll see a tremendous boost in speed. Regardless, most high-speed providers don't provide more than 30mpbs to the average customer. If you are one of the lucky few who gets the highest-tier Verizon you can get up to 300 mpbs per second.
So if you have the average high-speed internet provider, theoretically you can save downloads directly to the external hard drive if you are using atleast Hi-Speed USB, because you won't ever have the data coming in faster than 30mpbs and the slowest data transfer speed for newer connection types is 480mbps. But, if for some reason you have an archaic usb port in your computer and rocking a 12mpbs speed you might see some latency while directly downloading to the external.
Now, say you had the highest Verizon tier you could download and transfer just as fast to the computer as you would to the external harddrive with atleast Hi-Speed USB connection type or faster.
But lets have some fun and say you live in Kansas City and are one of few people that got Google's new service. You'll be seeing average speeds of 600 mbps per second. So if you were one of those peeps and wanted to save to an external harddrive and were downloading from an extremely fast server(They probably have to be located and on Google's figer optic as well haha), you would want to have at least a usb 3.0 or thunderbolt port/connection. So you could transfer the 600mpbs in real time.
Last but not least, be safe out there downloading
----
Sorry it's a boring friday at work, thought I'd take some of you to school