Bicycle Mechanics - Front derailleur dilemma

I have just replaced the crankset on my Mongoose ATB and have put the old one on the Hardrock. The problem is with the Alivio that I have put on the Hardrock. I am thinking that the derailleur that is on the the bike won't work with the crankset. The old crankset is an off-brand that has a big chainring with 48 teeth and the Alivio is 42t.

So, my question is this...

Do different cranksets require different derailleurs? And what is the difference between a top and bottom pull derailleur? I ask this because I do not know which one I would order if it turns out I need to get a new derailleur.

Thanks.

It should not. Try this. Lower the front der. about 2 to 4 mm above the outer ring. When you shift the front der. to the outer ring the outer cage on the der. should be about the above mentioned height.
A top pull der. is one that activates by the cable pulling from the top. The opposite applies on a bottom pull. Now there is something else. It is called a top pull bottom swing, and a top pull top swing. There is also a bottom pull top swing, and a bottom pull bottom swing. This is simply the location of where the cable bolts to on the front der.

You need to match the curvature of the outside edge of the mech, to the size of the big ring. There is some latitude, and you can adjust by moving the mech up and down.
You fit a top or bottom pull mech according to the cable routing on your frame. Top pull ones are for muddy conditions.

I don't think you'll need a new one so long as you can drop it down to just above the teeth of the new outer ring.