If you've already got a hotspot and your home is small enough for its WiFi to cover it, then yeah you only need the external antenna. As for the difference between the two, your hotspot is like a cell phone that communicates with the tower via LTE with one radio, and sets up a WiFi hotspot on a second radio. LTE to WiFi.
What the rebroadcasting device does is basically set up a cellular LTE tower inside your home. LTE to LTE. Your phone (or in this case your hotspot device) would communicate with the internal antenna in your home. The rebroadcasting unit would amplify its signals and send it over the external antenna to the Verizon tower. It's a better solution if you need multiple cellular devices to get LTE service in your home (e.g. you want your hotspot, your phone, and your wife's phone to all get LTE service). But if all you need is for the hotspot to get an LTE connection, then the hotspot is the better way to go. Verizon has some WiFi calling capability, so you could actually use that on your phone to make/receive phone calls through the hotspot. You don't necessarily need a cellular signal booster in your home.
https://www.verizonwireless.com/support/wifi-calling-faqs/
The frequency bands listed as supported by the antenna on the Amazon description cover Verizon's LTE bands (2, 4, 13) so the antenna should work provided the tower provides a 4G signal. I'd suggest finding the nearest Verizon tower to you. Then drive there with your hotspot (or phone if it supports Verizon LTE) to confirm that the tower delivers a 4G signal. If it doesn't, try the next closest tower. Repeat until you find one which provides a 4G signal.
https://www.wilsonamplifiers.com/blog/finding-cell-tower-locations-the-complete-guide/
That exercise will also give you the direction you need to aim the antenna. Yagis tend to be extremely sensitive to direction, so you're going to have to aim it very precisely. And knowing exactly where the tower is will go a long way to helping aim the antenna.
All three Verizon LTE bands are FDD, so there's no theoretical range limit (other than signal strength). However, you will probably have to deal with curvature of the earth limiting your line of sight distance. For example, if you mount the antenna 30 ft above the ground and the tower is 100 ft high, line of sight distance is only 22 miles. Further than that and the earth's curvature blocks the two antennas from seeing each other. So you'll first want to make sure it's even possible to get a signal at your home even with the best possible antenna. Otherwise you're going to have to look into building a tower on your property and mounting an antenna on top of it. (In which case finding a wireless Internet provider may actually be better.)
http://www.calculatoredge.com/electronics/lineofsight.htm
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