Broadsheet and tabloid describe roughly how big a newspaper is. They are not standardised measurements, and that is the first thing to get straight, because two printers can both sell you a broadsheet and hand you very different sheets of paper.
Newspaper Club's traditional broadsheet is 380 by 578mm. Its digital broadsheet, sold in the same catalogue, is 350 by 500mm. MakeMyNewspaper's broadsheet is 11 by 22 inches, which is 279 by 559mm, narrower than either British version and shaped quite differently. All three are correctly called broadsheets.
The real differences, beyond size
A tabloid is roughly half a broadsheet. That is the origin of both names: a broadsheet is the full press sheet and a tabloid is that sheet folded again. The practical consequences follow from the proportions rather than from the area.
A broadsheet is read folded. Nobody opens a full broadsheet out on a train. That is why traditional newspapers put the masthead, the lead story and one strong photograph in the top half of the front page, and it is a real design constraint rather than a convention. Your front page has to work when the bottom half is hidden.
A tabloid is read flat and handled easily. The whole front page is visible at once, which is why tabloid front pages tend towards one enormous headline and one image. It also posts more cheaply, stores more easily and does not require a table to read.
Column count is where layouts break
This is the practical difference that catches designers moving between the two. A five column grid on a tabloid gives a comfortable line length of around fifty millimetres. The same five columns spread across a broadsheet's 350mm printable width gives lines of about seventy millimetres, which at nine point is too long to read comfortably. Your eye loses its place returning to the start of the next line.
Broadsheets therefore run more columns, typically six or seven and often more. If you scale a tabloid layout up to broadsheet without changing the grid, the result reads badly even though nothing is technically wrong with it. Rebuild the grid rather than stretching it.
Page count ceilings differ sharply
Newspaper Club's traditional broadsheet tops out at twenty pages and runs in multiples of two. The tabloid formats run to sixty-four in multiples of four. That is a considerable difference and it decides the question on its own for long publications.
The multiples of two on the broadsheet is genuinely useful in the other direction, though. A six page paper is impossible on any tabloid format and perfectly ordinary on that broadsheet, which suits a short, substantial publication like a manifesto, an exhibition guide or a single issue special.
Which to choose
Choose broadsheet when the size itself is part of the message. A broadsheet has authority and occasion about it, which is why it suits wedding papers, exhibition guides, campaign publications and anything meant to be kept. Choose it also when your photographs deserve to be large, because a full page image on a broadsheet is genuinely striking in a way it cannot be on a smaller sheet.
Choose tabloid when it will be handed out, posted, read on public transport or kept in a bag. Choose it when you have more than twenty pages of content. Choose it when you are printing a small number of copies, because the digital tabloid is the cheapest short run option available. And choose it if you are working from stock templates, because far more of them are drawn to tabloid proportions.
If you are still undecided, the practical test is the word count rather than the aesthetics. Run your draft through the words to pages estimator at both sizes. If broadsheet saves you no pages, you are paying for a larger sheet you do not need.