Limewash: thin, tinted, breathable
Limewash is not paint. It is a mineral coating that sinks into masonry and cures as part of the wall, which is why it can soften brick without sealing it shut. All of the brick stays visible; the color is shifted rather than covered.
Standard exterior paint forms a film on top of brick. Moisture that gets behind it has nowhere to go, and eventually the film blisters off and takes face material with it. Mineral limewash bonds chemically with the masonry and stays vapor open, so the wall dries out rather than trapping water.
Fresh limewash can be worked back or washed off within a window after application, which is part of how the distressed look is achieved. That also makes it the more forgiving of the two decisions.
German Smear: thick, textured, permanent
German Smear is a mortar finish worked onto brick by hand so that some face brick shows through and some is covered. Done well it looks like it has been there a hundred years. Done badly it looks like someone dropped stucco on a wall.
Cured mortar on brick is not something you wash off, and removal is aggressive work that can damage the brick face. This is not a finish to try out.
The version that causes real trouble is a cement stucco skim, which seals the wall and traps moisture behind it. Ask any bidder specifically which they are applying.
Which one your brick suits
Most porous brick takes limewash well. Hard-fired brick, previously painted brick, and sealed masonry are the cases that need testing, and testing means an absorption test on your actual wall rather than a guess.
German Smear reads best on brick with visible texture and color variation, because the partially-covered face is the whole point. On a flat, uniform modern brick the effect can look applied rather than aged.
What to insist on before either
A real sample panel on an inconspicuous part of your own wall, at the coverage you want, viewed in your own light at different times of day. Photographs of other houses are not a substitute, and on a German Smear you should live with the panel for a day or two before deciding.
Ask which product line is being used by name. A true mineral product has a documented system behind it; a tinted latex sold as limewash does not.