If something happens consistently or someone does something consistently, it may not happen all the time, but we can depend on the result.
Something that happens constantly happens all the time, without letup or with few breaks.
"He consistently scores at the top of his class." The scores do not happen all the time, but when the do, he always (or almost always) achieves a high score.
"It rains constantly here during June". It rains almost all the time during June.
Note that I could turn this around and say:
"June is consistently rainy here." It does not mean the same thing as the first. It means that whenever June comes (which isn't all the time) we can depend on it to be rainy. The rain may not be constant during that time, but there will be a lot of it.