Berg vs. Burg
These two words show up abundantly in family and place names across the Germanic languages (German, Swedish, Dutch, English etc.), so it’s really helpful to keep them straight. Which is tricky, because in modern English, the two have come to be pronounced the same. They didn’t start that way, though, and in the other Germanic languages, they are still distinct.
A Berg “bairk” is a mountain, and a Burg “boork” is a defensive medieval castle. In English, we’ve merged the -er- and -ur- into just -r-, but in German the two are still fully distinct.
The long and short of it~
The short of it:
English calls all sorts of things castles. German has two separate words: Burg for a castle-castle, and Schloss for a palace-castle. Actually, German tosses in Palas [paa-LAAS] and Palais [paa-LEY] to be Frenchy-fancy about palaces. English simply translates them both as ‘castle’. However, a Burg is uncomfortable but very defense-oriented, and medieval, while a Schloss won’t resist more than a bandit raid, but it is much more comfortable, and they were largely built after the Middle Ages (and the introduction of cannons, when old-style Burgen became less useful). Since the Schlösser were generally built after the period when people adopted family names and founded cities, Burg, not Schloss, is the one that generally shows up in names.
The long of it:
Both come from an old word meaning ‘high place’. Berg means mountain; the English equivalent lives on only in the old meaning of ‘large grave mound’ as in Tolkien’s barrow wights in the Lord of the Rings. English did borrow German Eisberg for iceberg (‘ice mountain’), and the word shows up in several related languages meaning mountain (like Danish bjerg) or a high place, like Russian bereg or Polish brzeg ‘bank, shore’.
I’m pretty sure you aren’t going to have barrow or barrow-wights in your family name.
The variant burg had the meaning of a high place that is defendable. In German, it has kept the meaning of a medieval defensive castle, die Burg “boork”. Since towns often grew up around a lord’s castle, lots of German town names end in -burg, ditto people named for such towns. In the 1700’s, Britain needed a new king, and the next eligible heir was a Hanoverian German monarch, George I of England, so -burg was used for a number of Colonial settlements in indirect honor of his German background, notably Pittsburgh, named after Prime Minister Pitt who led Great Britain to victory in the French and Indian Wars.
But what about the unborrowed English original? Old English had burg/burh "boorkh" (English has almost completely lost that kh sound for fortified sites), and that reached modern English in several different forms: in the north and in Scotland it was burgh, and the -gh was pronounced as a kh sound. Today, the Scots generally pronounce burgh names like Edinburgh as Eddinbahdda. The term burgh is still a municipal designation in Scotland and northern England.
Central England developed old burg into borough (“boro”), which is still used as a particular class of municipality; in the U.S. it is generally a county that is part of a city, like the five boroughs of New York; London and Montreal have similar usage; Australia has largely abandoned the term by now.
Further south in Englad the plural byrig gives us place names in -bury, like Canterbury.
But what about -burger or -berger? In the Germanic languages, you’d take on -er to mean, in this context, ‘somebody from X’. Your ancestor moved from the village of Rothberg (‘red mountain’)? They get called the Rothberger, ’the guy from Red Mountain’. Someone who lived on Altburg gets called Altburger.
But Burgher, all by itself? That’s usually ‘the person from the borough, i.e. from the city’, and that carried special rights and privileges.
But Berger? That was usually ‘so-and-so from the Mountain and not so-and-so at the woods (Atwood)’. Though occasionally it’s an unrelated name from French berger (bair-zhair) meaning ‘shepherd’.
The word shows up in other languages: Italian borgo, Catalan burg, French bourg (many or all of these were borrowed from Germanic languages), and apparently even in Arabic burj ‘tower’.
The Slavic languages use a different word that originally meant compound or contained area, ghord-, which comes out as hrad in Czech for ‘castle’, but as (fortified) city in Russian gorod/-grad: Novgorod, Stalingrad etc. The same ancestral word came out as hortus in Latin, meaning garden (as in horticulture: a garden was a walled-in area), and German developed it into Garten (which gave them Kindergarten, which gives us Kindergarden ‘children’s garden’). It came out in English as yard. And remember Latin hortus? Old English borrowed the Late Latin form orto to mean fruit, and a fruit yard or orto-yeard turned into orchard.
Dutch gives us names like Vandenburg ‘he from-the-castle’; Sweden has names like Göteborg, the city of the ancient Götar/Geatish tribe, and the tennis star Björn Borg.
The main takeaway from all of this is: berg and burg are not simply different spellings but actually refer back to separate meanings. It might be tough to trace that far back in your family history, but hopefully this helps give you some clues in your search.