What's the difference between Machine Learning classifications in Computer Science and Statistics on arXiv?
The cs.LG category has a broader scope. As per arXiv's category taxonomy, the category descriptions are:

stat.ML (Machine Learning) -
Classification, Graphical Models, High Dimensional Inference

and

cs.LG (Machine Learning) -
Papers on all aspects of machine learning research (supervised, unsupervised, reinforcement learning, bandit problems, and so on) including also robustness, explanation, fairness, and methodology. cs.LG is also an appropriate primary category for applications of machine learning methods.

More specifically, in this page, an additional guideline for cs.LG is provided (emphasis mine):

Relationship to other categories: If the primary domain of the application is available as another category in arXiv and readers of that category would be the main audience, that category should be primary. Examples include applications to computer vision (cs.CV), natural language processing (cs.CL), speech recognition (eess.AS), information retrieval (cs.IR; includes document classification and topic modeling), crowdsourcing (cs.HC), quantitative finance (q-fin), and quantitative biology (q-bio). Papers discussing the foundations of neural network architectures (activation functions, spiking neurons, etc.) should list cs.NE as primary, as should papers applying biologically-inspired optimization techniques such as evolutionary methods. Papers working with the properties of specific signal types (e.g., sound, EEG, hyperspectral, ultrasound) should consider cs.SD (sound, including music), eess.IV (images and video), or eess.SP as primary. cs.LG is not appropriate for papers studying human learning such as computer-aided instruction, where cs.CY is a better fit. Papers categorized with cs.LG as primary are automatically cross-listed as stat.ML and vice versa.

So, based on the content of your paper, you can follow these guidelines to decide the appropriate category. For papers applying machine learning to other fields, such as computer vision, neither would be the primary category, but can be present in the cross-list.
However, if you need to select one of these two as the primary category, then as per the last line in the quoted paragraph above, no matter which one you choose, the other will always be listed in the cross-list. So, the paper will end up appearing in both categories.