I thought your base represented your centers of operation, your so called "survival" area. Bases can be designed to look like a house but dont have to look like one. If someone told me to build a base in other words, it would be more of a generalization than if someone told me to build a house.
I think, A base is generally more suited for pvp, or just everything in general, a base will, or rather, should contain multiple things like a farm, enchant room, storage, sleeping, crafting and smelting, where as a house will be made strictly to live in, while the appearance doesn't really define a base or house, houses are generally more attractive as they only have two focuses, living and looks, then again eye is in the beauty of the beholder.
A house is more of a smaller building that has nice decoration and fits all of your basic survival needs while a base may have more advanced items and it most commonly used in SMP. Bases may also be bigger and may most of the time be well-protected or hidden in Faction Servers.
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All houses can be bases, but not all bases are houses. In other words, if I told you to build a survival base, there are loads of different ways you might fit that description, since it was very vague. However if I told you to build a survival house, then you are building whatever your perspective of a mine craft house is.
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Bases are usually very large and consist of multiple buildings or areas and sometimes the enviornment around it.
A house, on the other hand, is not necessarily small but is probably going to be smaller than a base. A house is also a singular building.