Actually there is no diving place near Buenos Aires.
The nearest place is located in Mar del Plata (400 kms away from Buenos Aires) and conditions are pretty variable. The conditions are quite variable because the diving places are in open sea without any natural shelter, so tide, currents, weather and visibility can vary from incredible good to terrible bad from one day to another. Frecuent cancellations owing to poor conditions.
The next place could be "Cantera El Terrible", located 500 kms away from Buenos Aires. In fact it's located in Uruguay, near Salto city (Concordia city). It's a quarry. Visibility is between 3 to 6 meters and very little to see. Max depth is 12 meters. There is a sunken bus there as a wreck.
The next place could be "Cantera Ferrando". It's near Buenos Aires, just crossing the Rio de la Plata river in Colonia Uruguay. With the fast ship you can be there in two hours. It's also a small quarry. Visibility is between 3 to 6 meters. Max depth is 10 meters. I've been there two weeks ago and visibility was pretty bad. Colonia is an very nice city to visit anyway.
Going further, you can dive in "Las Grutas" (1.100 kms away from Buenos Aires). It's open sea, but a in a gulf, so waters are quiet and diving is always nice. There are 9 meters tide from low to high, so diving should consider this difference. Tide current should be considered. There is a wreck, a 25 meters fishing cutter sunk for diving (20 meters depth in high tide). Water temperatures range from 20 to 10 degrees celcius from summer to winter (we are entering winter now).
The most notorius sea diving location in Argentina is "Puerto Madryn" (1.500 kms away from Buenos Aires). There there are several wrecks to visit, diving with sea lions and crystal waters. As it's located well south of Argentina, dry suit is used year round if you plan repetitive dives.
Some other places to mention are :
"Embalse Rio Tercero - Segunda Usina - Córdoba" (800 kms away from Buenos Aires). It´s a dam, fresh water. Nice diving though little to see. Conditions vary with wind.
"Embalse La Florida - San Luis" (1.000 kms away from Buenos Aires). It´s a dam, fresh water. I've not dived there, so I cannot add much.
South lakes (starting 1.800 kms away from Buenos Aires). They are lakes in the mountains. Incredible beautiful enviroment. Crystal waters from melted snow. Terrible cold, dry suit year round, altitude diving. Excellent visibility though little to see. Only to mention there is a sunken wood as a consecuence of a landslide.
Finally Antartica. Too far away and this should be treated as a special journey.
Consider that there is not so many flights to connect the diving places, so you should use the bus service to go. This is time consuming.
Finally normally the dive operators do not speak english, so arranging a dive excursion if you cannot speak spanish could be a nightmare.