In most cases, the choice comes down to preference. Some people like the notes readily available on the same page, while others prefer them all in one location. If you are writing for a university or college course requiring the use of Chicago style, ask the teacher for his preference. If you plan to submit your writing to a journal or magazine, check a past issue or query their editing department for their preferred note placement.
The strongest argument in favor of footnotes is that they are readily available to the reader, while endnotes force the reader to flip to the end of the text. According to the style manual, in the interest of preserving the flow of the text, footnotes should generally be used before endnotes. However, numerous lengthy footnotes can be very difficult to integrate into a text without distorting its presentation. At its worst, "footnote bloat" can take up most or even all of a page.
The endnote format is more appropriate for texts with very extensive notes, especially if the notes contain charts, tables, graphs or other imagery. Using endnotes should be avoided if the notes are very short or repetitive, because constant page turning can frustrate the reader. By the same token, endnotes of any length force the reader to actually locate the endnote in question, which can badly disrupt his reading of the text.
In long texts with extensive sources and notes, it is acceptable in Chicago style to separate citation notes -- short notes containing only source information -- from lengthier notes. In this case, the citation notes should be numbered as usual and placed as endnotes. Longer notes should be indicated by symbols and placed as footnotes. While this may contradict the manual's earlier advice to use endnotes for longer notes, it is done this way because a list of endnotes can get quite long, making the use of symbols rather than sequential numbering impractical.