MobileNet vs SqueezeNet vs ResNet50 vs Inception v3 vs VGG16
The models Apple makes available are just for simple demo purposes. Most of the time, these models are not sufficient for use in your own app.
The models on Apple's download page are trained for a very specific purpose: image classification on the ImageNet dataset. This means they can take an image and tell you what the "main" object is in the image, but only if it's one of the 1,000 categories from the ImageNet dataset.
Usually, this is not what you want to do in your own apps. If your app wants to do image classification, typically you want to train a model on your own categories (like food or cars or whatever). In that case you can take something like Inception-v3 (the original, not the Core ML version) and re-train it on your own data. That gives you a new model, which you then need to convert to Core ML again.
If your app wants to do something other than image classification, you can use these pretrained models as "feature extractors" in a larger neural network structure. But again this involves training your own model (usually from scratch) and then converting the result to Core ML.
So only in a very specific use case -- image classification using the 1,000 ImageNet categories -- are these Apple-provided models useful to your app.
If you do want to use any of these models, the difference between them is speed vs. accuracy. The smaller models are fastest but also least accurate. (In my opinion, VGG16 shouldn't be used on mobile. It's just too big and it's no more accurate than Inception or even MobileNet.)
SqueezeNets are fully convolutional and use Fire modules which have a squeeze layer of 1x1 convolutions which vastly decreases parameters as it can restrict the number of input channels each layer. This makes SqueezeNets extremely low latency, in addition to the fact they don't have dense layers.
MobileNets utilise depth-wise separable convolutions, very similar to inception towers in inception. These also reduce the number of a parameters and hence latency. MobileNets also have useful model-shrinking parameters than you can call before training to make it exact size you want. The Keras implementation can use ImageNet pre-trained weights too.
The other models are very deep, large models. The reduced number of parameters / style of convolution is not used for low latency but just for the ability to train very deep models, essentially. ResNet introduced residual connections between layers which were originally believed to be key in training very deep models. These aren't seen in the previously mentioned low latency models.