After a prolonged and deep solar minimum at the end of Cycle 23, the current Solar Cycle 24 is one of the lowest cycles. These two periods of deep minimum and mini maximum are separated by a period of increasing solar activity. We study the cosmic-ray intensity variation in relation with the solar activity, heliospheric plasma and field parameters, including the heliospheric current sheet, during these three periods (phases) of different activity level and nature: (a) a deep minimum, (b) an increasing activity period and (c) a mini maximum. We use neutron monitor data from stations located around the globe to study the rigidity dependence on modulation during the two extremes, i.e., minimum and maximum. We also study the time lag between the cosmic-ray intensity and various solar and interplanetary parameters separately during the three activity phases. We also analyze the role of various parameters, including the current sheet tilt, in modulating the cosmic-ray intensity during the three different phases. Their relative importance and the implications of our results are also discussed.

23 pages include 13 figures and 6 tables. Accepted to Solar Physics (2015)