This is not the best translator but google translates them both as sandwich and sandwiches respectively. As I was typing it in "boter" first came up as "butter", so I checked it out separately and "ham" is "ham". "butter" + "ham" doesn't seem like sliced bread to me, but the makings of a sandwich. "brood" is bread, and a diminutive ending has been added to it. Interesting!'
Aahhhh! This dictionary does have "boterham" with both meanings "sandwich" and also "piece of bread" or "slice of bread".
"broodje" is listed as a "bread roll" or "bun" and gives an example of herring on a roll as "broodje herring" which certainly sounds like another kind of sandwich. I had one exercise in a lesson (maybe it was in the English from Dutch course? I stumbled on that one as well.) that gave "sandwich" as the dutch word and asked for the English word which of course was "sandwich", but I don't know if that was an error. In English we have many words for sandwich (a panini, a sub....); so of course, there can be more than one in Dutch.
The Dutch REALLY love bread so, naturally, we have different words for it. A slice of bread is 'een boterham' and if you put, say, cheese on it, it is 'een boterham met kaas'. If you're with someone at your house and you're hungry, you can tell them tham you are going to make yourself 'een boterham' which doesn't mean you're going to slice a loaf of bread, but you're going to take a slice of bread and put something on it. If you take two slices and put cheese between them, it's 'een dubbele boterham met kaas'. When talking about one or two slices of bread with something on it, we sometimes refer to that as 'een sandwich'. Also, if the cheese is on a bun, it's called "een broodje kaas" and a bun with multiple toppings (I think in the US they would call that a sub) is usually calles "een belegd broodje". And there's much much more types of bread...