Both types of martingales aim to address resistance by keeping the horse's head and neck in a position where the bit and your aids are more effective.
The standing martingale really just keeps the horse from raising his head above certain height to avoid the bit. However, smart and lazy horses learn to lean or balance themselves on the martingale. I use them rarely and for a short term, rather than long term fix.
Running martingales are more versatile. When adjusted correctly, they put downward pressure on the bars of the horses's mouth when he raises his head above a certain level and release that pressure when the horse drops his head. So used in moderation they can be an effective tool.
One big caveat - I almost *never* see them adjusted correctly. Too loose, they're ineffectual, and too tight, they put constant pressure on the bars of the horses mouth, which is contrary to good training and really abusive. If you use one, make sure someone who knows what they're doing adjusts it, and evaluate the adjustment each time you ride with it.