A two-part answer.
One. Regiments and Brigades are the same thing, a third of a division or three battalions. Tactical regiments exist in many services. A seperate armor birgade is called a regiment in the US Army (an armored cav regiment).
All brigades are called regiments in the American Marines. (Their seperate regiments are often called brigades) When you see a report from a journalist enbedded with the Third Battalion of the Eighth Marines, that is the Eighth Marine Regiment.
So in a tactical sense, they are more or less the same thing.
Now in an historic sense, the US Army has screwed its regmental system to a fare-thee-well. In olden times, soldiers served their professional lives in a single regiment. Regiments developed an esprit de corps (and more than a few eccentrities).
In WWII, for example, a division would have three tactical brigades, each of which was usually a single infantry regiment plus supporting troops.
That went away in the 1950 when the (US) Army went to a funny “Pentaomic” Division, which had five “battle groups,” smalled than a brigade/Regiment but bigger than a battalion.
As a result the regimental system was rethunk. The idea was a soldier would serve with a battalion of (say) the Seventh US Infantry Regiment (the Cottonbalers) in Germany and then rotate back to the US to serve in another battalion of the Seventh.
The two battalions would not have anything to do with each other except they would share a regimental designation. They would be in seperate divisions and corps. The regimental system became a fraud.
Added to this is the fact that modern soldiers like to move around. After a tour in Germany they want to go to Hawaii or Korea. Such mobility wrecks the idea of a regimental home.
As a result the (US) Army has a hodgepodge of battalions scattered willy-nilly all over the world with all sorts of regimental designations. Each unit has a regimental color with streamers for the battales found by the unit’s forebearers. The system tries to keep up the old traditions and prid with mixed sucess.
In the British military the problem has been a seemingly-endless series of consolidations.
The old Artist’s Rifles (raised in WWI) has gained a number (22) and is now the Special Air Service. The county regiments have been consolidated (as have the counties themselves). Still vestiages of the ol days remain.
When you see a British soldier wearing a hat that make you ask “Where did he buy that?” He is wearing the distinctive headgear of an old and proud regiment. Since the British Army is smaller now that it used to be, some or even many of the regiments are down to a single tactical battalion.
So the 43rd Foot might well be just a single battalion that serves in a brigade of the First British Armoured Division.
OK, get all that? Aren’t you sorry you asked?