The short answer
The words on gin labels refer to very different things: the production method, a flavor tradition, or the alcohol strength. London Dry, Old Tom, and Navy Strength aren't a ranking of quality.

London Dry and Distilled Gin
With distilled gin, the alcohol is redistilled together with botanical flavorings. London Gin follows additional rules about what can be added afterward. London Dry doesn't have to come from London. The term describes a method of production, not a flavor.
The difference lies in what happens afterward. With distilled gin, flavorings can still be added after distillation; with London Gin, almost nothing can. The flavor there has to come from the distillation itself. On the shelf, this means London Dry promises a method, not a specific taste.
Old Tom and the modern styles
Old Tom harks back to older, generally somewhat sweeter gins. How sweet is up to the producer. New Western or Contemporary Gin are style descriptions: other herbs and spices step forward alongside the juniper. Read the product description rather than judging the flavor by the name.
New Western is more of an approach than a rule. Juniper is still recognizable, but it no longer leads. If you got to know gin through a juniper-forward classic, these bottles can sometimes feel too restrained — and vice versa. That's a matter of taste, not quality.

What Navy Strength means
Navy Strength means more alcohol, often around 57 percent. There's no fixed recipe behind it. Check the number on the label. In a cocktail, a stronger gin shifts the balance — you can't simply dose it the same way as a lighter one.
The higher strength changes more than just the effect. Alcohol carries aroma compounds, so a robust gin comes through more clearly in a drink with lots of ingredients. Neat, a few drops of water are worthwhile: they take the edge off and let the herbs come forward.
Sloe Gin and Pink Gin are something else entirely
Sloe Gin is a gin-based sloe liqueur and tastes in a completely different direction. Pink Gin can be a cocktail of gin and bitters, or a pink-colored product. Check the color, sweetness, and category separately. When shopping, asking about your planned drink helps more than searching for the "best" style.
This matters for recipes. If you swap dry gin for Sloe Gin in a Martini, you don't get a variation — you get a different drink: sweeter, fruitier, weaker. If a recipe just says "gin," it almost always means dry gin.
Which label detail matters when
Three different things appear on the bottle, and they can't be weighed against one another: the production method (London Dry, Distilled), the flavor tradition (Old Tom, New Western), and the strength (Navy Strength). One bottle can carry several of these at once. A London Dry at Navy Strength isn't a contradiction.
This order helps when shopping: first decide the purpose (neat, tonic, cocktail), then the flavor direction, then the rest. Starting with the categories means sorting terms instead of drinks.