Imagine a higher-order factor model with three lower-order factors defined by the items and one higher-order factor defined by the three lower-order factors. Now imagine a bifactor model with one general factor defined by all the items and three smaller factors defined by subsections of the items. The bifactor model typically requires the smaller factors are set as orthogonal to the general factor. This way, you might think of the smaller factors as residual factors or as method effects. But with the higher-order model, the higher-order factor is explicitly related to the lower-order factors—it’s defined by them. In this way, the lower-order factors are like subcomponents or facets of the higher-order factor. Brown talks a bit about these models in his CFA text (https://www.guilford.com/books/Confirmatory-Factor-Analysis-for-Applied-Research/Timothy-Brown/9781462515363). Little also addressed them in his longitudinal SEM text (https://www.guilford.com/books/Longitudinal-Structural-Equation-Modeling/Todd-Little/9781462510160).

Imagine a higher-order factor model with three lower-order factors defined by the items and one higher-order factor defined by the three lower-order factors. Now imagine a bifactor model with one general factor defined by all the items and three smaller factors defined by subsections of the items. The bifactor model typically requires the smaller factors are set as orthogonal to the general factor. This way, you might think of the smaller factors as residual factors or as method effects. But with the higher-order model, the higher-order factor is explicitly related to the lower-order factors—it’s defined by them. In this way, the lower-order factors are like subcomponents or facets of the higher-order factor. Brown talks a bit about these models in his CFA text (https://www.guilford.com/books/Confirmatory-Factor-Analysis-for-Applied-Research/Timothy-Brown/9781462515363). Little also addressed them in his longitudinal SEM text (https://www.guilford.com/books/Longitudinal-Structural-Equation-Modeling/Todd-Little/9781462510160).

The models are different, so it depends what you are hypothesizing. Bi-factor has the general factor influencing all the manifest variables (and I'll just be referring to simple models without additional structure) by second order has the higher order factor influencing each of the first order factors. Since there are usually fewer first order factors than manifest variables this model has fewer degrees of freedom. These two (and the correlated factor model) will each produce similar fits in many situations. Often the choice between them should be informed by which makes the most sense and which will be the most useful.

ps. I was typing mine when Solomon's very nice answer must have come up. I am repeating part of what he said, so I guess hearing it twice is okay. I guess I better add something ... https://www.researchgate.net/publication/283760085_John_Carroll's_Views_on_Intelligence_Bi-Factor_vs_Higher-Order_Models includes some good plots of the two models and discussion. Also, when I've compared these I used lavaan and I think the examples (maybe on GitHub) included these.

Article John Carroll’s Views on Intelligence: Bi-Factor vs. Higher-O...

 Thanks a lot for your valuable answers. thanks for article. Beujean a known structural equation modeling scholar. I will read this article.