What's the difference between "at an increased rate" and "at an increasing rate". It seems to me both of these are correct and no difference. Am I right?
"at an increasing rate" means that the rate itself is increasing; hence a statement about the behaviour of the rate at the point of observation. It is therefore more a description of the rate in terms of continuous behaviour.
"at an increased rate" means that the rate has increased relative to some prior value. It therefore merely compares two specific values and states that one exceeds the other. It is hence a description of the rate in terms of discrete observations.
As an aside, I'd note that confusion between the two is very common; typically we observe some specific values of a sequence, infer a trend and turn that into a statement as to what is happening now.
If you think about it, the discrete version is a factual statement about actual observations, whereas the continuous one is often just an inference.
For most purposes, no one is going to care as to the difference. In some contexts, such as an academic paper, the distinction it important.
Increased as a past participle merely means augmented relative to some prior value, e.g., a car traveling at 20 mph that was previously going at 10 mph. Increasing means that the rate has been going up, and continues to go up. This would be a car that is in the process of accelerating, e.g., it is going 20 mph now, was going 10 mph a minute ago, and will be going 30 mph shortly.
If you like, the only concrete difference is that the increasing rate will go up in the future, while an increased rate makes no claim on the future.