That really depends on what level of physics you are dealing with. Even physicists don't normally care too much about the difference, so we tend to use the terms interchangeably.
However, if you like, magnetic flux often refers to the
B field. And yes, it is also called magnetic flux density. The word density should automatically tell you that we are dealing with a certain amount in a unit space (volume, area, etc.). In this case it is the area we are dealing with, hence the reason why total flux is equal to the magnetic flux density times the area it is passing through. Technically speaking, it is a dot product between the two, since both are vectors.
As for the magnetic field strength, you are now talking about the H field. We usually talk about the
H field when dealing with materials. To put it in a simple picture, the field strength refers to how easy/hard it is for the magnet to alter the properties of said material. This is why we define
where

is the magnetic permeability - a constant related to the properties of the material in question. I have ignored the magnetisation and so on, but they're irrelevant to the discussion.