This is a fairly open-ended question, so perhaps some examples of various possibilities might be helpful. Any group equipped with the discrete topology is clearly a topological group. However, that topology would definitely not be very useful in most circumstances. If we have two different topologies on a group that can both be used to turn it into a topological group, I could see some scenarios in which it might be desirable to work with one rather than the other (for instance, perhaps one is compact/locally compact while the other is not; or perhaps one is coarser and it would be more useful to have "less" open sets in some study). What are some more scenarios/properties that might make one topology more "desirable" than another when we have various options for how to topologize a group?
Added from a comment: A priori, there would typically be few choices for how to topologize a particular group that makes it a topological group, as the product and inverse maps would need to be continuous. Sometimes it might be very difficult to even find more than one non-trivial natural choice for such a topology. In those cases, if we do have two non-comparable candidates (so one is not coarser than the other) for a natural topology, it might be useful to think about what to look for in a topology in order to get more "interesting" consequences.
The most common scenario I know of when we have two useful topologies on the same group is in functional analysis, where when given a Hilbert space $H$ (which is just a topological group, with some extra multiplicative structure...) we can use the strong or weak topology on $H$. The strong topology is the metrizable topology induced by the norm of $H.$ The weak topology is a different topology.
The most obvious question to ask is why on Earth anyone would use a topology other than the topology induced by the norm, which seems to be the most natural topology on the space. The reason is that, in infinite dimensions, the norm topology is poorly behaved: Not enough sets are compact! In the weak topology on a Hilbert space (or more generally, the weak* topology on a Banach space), the closed unit ball in compact. In an infinite-dimensional Hilbert space, this is always false for the norm topology. It is incredibly useful to have compact sets, since compact sets are what let you deduce that functionals attain their maxima and minima. It is very useful in the field of PDEs, for instance, to know that certain optimization problems have solutions, even though you're optimizing over a set which isn't compact in the norm.