The short answer is it depends, or use a bit of both.
Now days I don't think you even really do textures on objects, instead you do 'materials' or 'effects' on objects.
A material can have multiple textures (1 or more basic image textures possibly with transparency, another for normals/bump mapping, another for specular reflections (The shiny part of metal will reflect light more than the scratched dirty bits) another texture for parts that should 'glow'. Another for ambient light (although that will often be the glow texture). They will have inputs (so you can swap between multiple different textures for the 1 material, modify the colour (maybe depending on the team the model is on), change animation speeds.
You're likely to use bother depending on what kind of object you are dealing with. And you can also do hybrids between 1 texture over the whole object and individual textures and so on.
If you are talking about limited hardware (mobile/web). Or a game with simpler graphics (although in that case optimizing is less important then anyway), I think it really depends.
Firstly, how close are you going to get to the object. Something like a house will need multiple textures if you are going to be walking up close to the walls, standing on the ceiling etc... but if it's a top down game where the house never gets closer than a specific distance then it's fine to use 1 texture. Maybe you use both methods depending on which LOD model is loaded.
It will depend on the kinds of objects you are going to be using. Some more 'organic' things can only really be done with 1 big texture (think of a tree trunk). Of course something like a face will
It will also depend on how many objects you are going to be seeing on the screen at once. If you have a RTS with armies of people vs a smaller number.
For the point of view of which is easiest to implement programmer wise, there about the same. In fact if you implement the multiple texture system, that will implement the single texture option.
It should also be possible for you to use multiple different materials/textures on one continuous mesh via extra vertex attributes. One of those can be a texture id. This is much easier on modern OpenGL where you can share uniform variables between shaders (I don't know about DirectX but I assume it's possible there too). Of course if you have different properties for your shaders then this will be harder. It also takes more memory. And you have to make the meshes have that metadata (although that can be done automatically).
It's also about the content creation pipeline rather than the programming side of things. Are you (or your artist) happier making a model, unwrapping the mesh then paining on it. This will mean each model is individually textured and unique, kind of hand crafted. A Blizzard game might do something like this, it allows fast creation of simple content, that's low performance, just knock together single mesh models then paint on them with a tablet. They use a simple cartoonish painting style.
Or are you likely to share textures between various models. A game that uses tiles might want the stone brick texture on 90% of them but mix in a few extra bits on most of them. And unwrapping tile objects is going to be harder in may ways due to the geometry.