These are two different ways for software (e.g. UpdraftPlus) to indicate to an S3-compatible storage provider which resource is being requested. Amazon S3 supports both (path-style is considered legacy, but is still widely used, so bucket-style is preferred and is the default). So, a fully Amazon-compatible S3 storage provider will allow either. However, in practice, some only support one; hence the option is provided in case you need to enforce use of one rather than the other.
Posted in: Backing up, Configuration