Emma Goldsmith 

You are right in that Edge would be deemed a more secure way of using NMT as the service that manages it would be installed on client side.
Were as Language Cloud services are managed  via RWS. 

So far as feature gaps between the two, they are minimal.
I am not sure which documentation you are referring to but here it outlines the only difference being Language Weaver supports "quality estimations".
Working with dictionaries would be supported in both.

Does this clarify the differences or better yet - how alike these 2 services are?

Lyds

Thank you, Lydia Simplicio 
I selected Language Weaver Edge and now realise I need to set it up on my own server (I think), which seems complicated. 
Anyway, thanks for the explanation. 

Emma Goldsmith correct. Edge is an NMT service between client server (client side) and Studio and therefore the client would have more governance over how the data is being transferred. Whereas for Cloud (aka the average user) RWS host and the data is being transferred over the internet and we take responsibility for encryption and data security. 

Emma Goldsmith correct. Edge is an NMT service between client server (client side) and Studio and therefore the client would have more governance over how the data is being transferred. Whereas for Cloud (aka the average user) RWS host and the data is being transferred over the internet and we take responsibility for encryption and data security.