"The Custom House" is an autobiographical addendum that Nathaniel Hawthorne included as the introduction toThe Scarlet Letterafter he felt that the novel itself was a bit too short for publication. It also serves as a wonderful foundation that gives a nostalgic atmosphere to the re-telling of the story of Hester Prynne, and her vicissitudes.
In "The Custom House", Hawthorne also explains the rationale behind writing about his experience in the Salem Custom House is because he had already published a story of a very similar style about his experience at an Old Manse.
The Old Manse to which Hawthorne refers is a historically-famous dwelling located in Concord, Massachusetts, which was built in the 1770's by Ralph Waldo Emerson's grandfather, the Reverend William Emerson.
This place held a lot of sentimental value to Hawthorne because he lived there with his new bride Sophia Peabody for around three years. Those seem to have been particularly happy times for the couple, motivating him to publish in 1846 a collection of short stories titledMosses from an Old Manse.
Hence, to specifically address your inquiry, The Old Manse and the Custom House are completely different places; they are places to which Hawthorne had lived and worked at respectively, and he mentions them both in the autobiographical introduction ofThe Scarlet Letterjust as a clever stylistic device.