They might sound the same but there is a difference!
Working Out
No overall plan or structure
Focus on one workout at a time even if there are many in a week
General or non-specific improvements to fitness and skill
Quality of a workout rated by calorie burn or sweatiness
May include a wide range of different types of workouts depending on personal preference
Exercise for its own sake, or just for fun!
Motivation, or lack of, plays more of role
Training
Preparing for a specific activity, usually a competition or event
Planned, progressive, periodised (meaning main focus will shift through different phases of training)
Using stats from individual training sessions to assess how performance is progressing and find specific areas to improve on
Likely to include some cross-training but specifically chosen to support the main training focus
More focus on discipline, mindset and mental strength which may form part of the training
There is of course some overlap between the two. An athlete in training would hopefully still find a lot of what they do fun (otherwise it will get really hard really quickly!) but will rely more on discipline than motivation to get started. Someone mainly exercising for health and fitness may still put the effort in to improve in a particular area, their 5k time for example, but in a less structured way or without a specific race/event in mind.
It depends on your individual goals which one would suit you best. You might go through a structured training phase for an event then relax and freestyle it more.
For those training phases or to help reach a particular goal it can be helpful to work with a coach, someone who can provide a structured plan, give feedback and support all the way through the process.