What are the exact reasons people visit a Banya, now that everybody has shower/bath at home?
Okay, let's compare.

Shower:
- You wash away the contaminants and get your body clean.
- You turn it hot then cold and back to hot to make a contrast shower, which is quite positive for your health, particularly for your cardiovascular system.
- You have a kind of massage with the streams of water.

Bath:
- You wash away the contaminants and get your body clean.
- You can lie in the water for an hour and soak your skin so the old dead skin particles get peeled away when you rub the body with a sponge.

Banya:
- You come in and just lie on the bench, wait for a while. The atmosphere in Banya is hot and humid. The humidity is about 50-60 percent which is good for your skin compared to sauna because it doesn't dry it out as sauna does, increasing the tendency for premature ageing. Banya doesn't do that. It moistens your skin with humid air and the aromatic oils or something else (like medicinal herbs, lavender or pine-tree buds) you may add to the boiler. Then you start sweating. Sweating doesn't go as rapidly as in sauna because of the moistened air. Throughout the process your skin pores become clean and "breathing". When you come out of the banya you feel like you're born anew. The temperature of the skin and the muscles raises (the air temperature is also about 50-60 degrees Celcius) as well as the metabolism does. So the fat starts "burning away" because of this metabolism. Also it kinda rinses the skin and the blood vessels. So your vessels become clean washed through with this blood pump. The effect gets to a maximum in the steam area (парная). You lie on another bench, splash some water (additives are recommended) on the hot stones lying near the smoke discharge outlet of the boiler, which raises both the temperature and the humidity. Then you start lashing youself (gently!!!) with this besom made of thin birch twigs with leaves. You should steam it out in advance in order to soften the twigs. Basicaly you just put it in hot water in about 5 minutes before you start lashing. So you're lashing yourself gently (or someone else can help you) -- yes, it's what you thought about, it can be your man or just another woman -- the mechanical action is boosting even further what I said earlier about the effects on the skin and stuff. Then you just relax and lie calm for a while. Then you prepare water for washing, and do the washing, of course. Just like you do it in the shower. Oh I forgot. To achieve a similar effect like #2 for a shower when the lashing ends you can just run out of the banya and jump into a snowbank nearby or just spill a bucket of cold water over yourself. It's like tempering, you know. But the effect is much stronger than a simple contrast shower.
To sum it all up, banya provides you with all the positive stuff you can get from a shower, bath and a sauna at once and adds some more