The picture shows honey, table sugar, a glucose molecule and a fructose molecule.
Tl;dr- Swapping table sugar for honey in your tea or coffee probably won't make a big difference, rather focus on maintaining a healthy body weight by matching calorie intake with energy needs. Meet these requirements mostly through whole, minimally refined/processed foods that you enjoy. And of course incorporate some form of exercise into your life. These two things play a far greater role in disease prevention.
Honey is converted to glucose in the body the same way table sugar is. It's then either burnt, stored as glycogen or converted to fat(under rare circumstances). This would be the same for any other "natural" sugars.
Sugar is a generic term for any short-chain, carbohydrate, and contains 4 calories per gram just like any other digestible carbohydrate.
Some health organisations suggest limiting added sugars to 10% of total calories. Note this is not intrinsic(naturally occurring) sugars such as lactose in milk or sucrose in fruit.
The average person consumes 2000 calories, so 10% from added sugars is 50g. This limit could be a challenge for most people and maintaining this restriction is unrealistic.
In a 6 week weight loss study, one group consumed 43% of total calories from sucrose and the other group 4% total calories from sucrose. They found no significant differences in weight loss between groups, also no significant difference in blood lipids or fasting glucose.
Consuming an arbitrary amount of sugar does not cause diabetes, it is well known in the scientific community now that excess body fatness, physical inactivity, and genetics are the major causes.
The main negative associated with foods high in added sugars is that they are usually low in vitamins, minerals, fibre and phytonutrients.