I have created a dissolved buffer of 150 feet around a points from dataset A and want to know how many instances where points in dataset B fall inside that buffer. So, I did a "Join data from another layer based on spatial location" to get a sum of 123 instances where points of B are within 150 feet of a point in A. I then used the Near tool to calculate the nearest point in dataset A to each point in dataset B, limiting the search radius to 150 feet. I only get 94 instances where the nearest A feature is within 150 feet of B. I thought that maybe this was due to working in a GCS and switched to the State Plane for my region and got the same outcome. Below demonstrates a location where the blue points (B) within the buffer are counted but the purple points (B) are the only ones found using Near with a search radius the same as the buffer.
Reading the help for the Near tool says it "Calculates distance and additional proximity information between the input features and the closest feature in another layer". The keyword here is closest. Closest is different to the question of "find me all points within 150 feet". So the closest points are those to purple points.
Looking at your image your buffer polygon has a slight elongation to it which suggests to me that was created using GCS. I would start again. project your points A and B into your state plane, then do the dissolved buffer.
The elongation you see is likely just due to dissolving (merging) the buffers around each of the small points in the middle. I did try projecting everything and redoing and got the same result. I am trying to locate the closest small dot to each of the blue dots as long as that small dot isn't more than 150 feet away, so each blue dot should have a small dot within 150 feet since it is within the 150 foot buffer of a small dot. However only the (slightly smaller) purple dots are found to have a small dot within 150 feet using Near.
I am not sure if I understand this right, but it seems to me that you should invert the relationship in running the Near tool - i.e. not finding "Near X to Y", but findint "Near Y to X".