I think the two expressions are largely interchangeable, but I'm not sure your example sentences are the best use of these phrases.
As you said, to mothball (or, to put into mothballs) is a figurative expression meaning to put something away for a long time, as in storage. I believe, though, that to mothball means more than simply stow something away; it usually implies you get something ready for long storage, so that it might be usable again in a time of need.
Also, we wouldn't mothball something in order to buy a new one, but we might mothball something as a result of buying a new one. (In other words, because we've bought something new, we can put the old one into mothballs; that way, if our new one fails for some reason, we still might be able to bring the old one back into use.)
So, addressing both of these issues, I'd propose these two sentences for example usages:
Brian mothballed his old lawnmower after he bought a new riding mower.
Brian put his old lawnmower in mothballs after he bought a new one.
By putting the lawnmower into mothballs (figuratively, of course), I would assume that he did a little bit more than stow the old mower in the toolshed. For example, he probably made sure that all the gas had been run out of the fuel system. He might have cleaned under the deck very thoroughly, to prevent trapped moisture from rusting the deck. He might have even put some extra lubricant in the cylinder, so that the piston wouldn't get stuck after long months of disuse.
As I mentioned in my comment, I was able to find the expression used in a myriad of contexts, but it seems like a more commonly used term in the navy, as a significant number of the references I found seemed to be linked to ships or shipyards.
I did like this quote, though:
Life is too interesting to be put in mothballs.