Leash aggression is a very common problem. Two out of my three dogs have it; although with Jessie it is not such a big change in her behavior when she is on leash. Many people complain that their dogs display so aggressively that they are embarrassed to walk their dog. It is uncomfortable when your dog is acting like they are trying to kill another dog. Let alone trying to convince the other guardian that your dog is indeed friendly. Yep; been there.
So why do dogs react differently off leash versus on? There are many things that factor into the change that occurs in their behavior. Communication is the biggy; our dogs loose their ability to communicate freely when they are on a leash. When dogs approach each other off leash they are communicating the entire time. If you go to a park or dog beach where they can run freely and watch carefully you will see it. Restrained by a leash this cannot happen so anxiety rises in some dogs.
Communication is also messed up by a leash. A dog that may want to approach your dog may strain at their leash making them go up onto their toes; which is a dominant gesture. They hadn't meant to give off this message but the leash is causing it. Being restrained causes the "let me at him" message. Whining, barking, jumping and growling can all be caused by not having the freedom to approach.
A protectiveness can kick into gear as well; not with all dogs but many. If a dog is off leash they tend to move away from their guardian. But when they are very close to their guardian they tend to be a little protective over their one and only. My dogs do it; when I was at the dog beach I was petting a very cute little dog. Luke immediately came over to make sure that this dog knew that I was "his" person. He wasn't aggressive about it; just pushy.
And the real issue to deal with is the humans, yep we make matters worse by our reactive behavior. Pulling on the lead and getting all tensed up tells our dogs that this is a stressful situation. So what may start out as a small issue snowballs into a monster very quickly. You need to relax; you are not alone. So many people have the same problem. Just yesterday I passed a couple with two very cute Boston terriers who were displaying to Jessie like mad. What a ton of noise; and Jessie is not one to take a challenge lightly, nope. So I just smiled as we passed; asked Jessie to chill out and she did just that. Had I reeled her in close to me; things would have gotten much worse.
Bring treats on your walk; teach your dog that walking past other dogs is a great thing. Luke use to be very reactive to walking past other dogs; now it is a rare occurance. Luke loves to catch so when we would see another dog I would immediately relax and get him to catch treats. Engaging your dogs helps them and you; it is hard to two things at once so if you are already involved in an activity it is much less likely that either of you will be able to do things like stress about another dog. No tugging on the leash, no change in pace; nothing to make him feel that this was a bad situation. Your behavior makes a huge impact on your dogs behavior.
Leash aggression is a fallout of our leashed society. Of course we cannot have dogs running everywhere; but we can lessen the blow of being on leash by relaxing. Lead by example; chill out.