Both systems are hierarchical; but LCC starts with leading letters while DDC is constrained by 9 uber-digits to start the bins rolling. There is, of course, some overlap because in 19thC there was a concept called Geography, almost a tangible thing, and everyone was happy about where geography ended and history or science began. But DDC treats Geography 910 as a subset of History in the 900s; while LCC bins it in the Gs with Anthropology and Recreation. I'm a lot less certain about the boundaries of geography than Putnam or Dewey: where does biogeography stop being science?; heck where do you put ecology? Clearly there wasn't much need to allocate library shelving to space flight in 1902, except to put copies of novels by Jules Verne and HG Wells. Now, libraries need to find room for such material, not only on shelves but also in the catalogue. Squeezing a lot of completely new post-19thC information in can mean dauntingly long accession numbers, which have to be put on the back of each relevant book. This problem is more pronounced in Dewey because their Universe was set by the 9 leading numbers in 1876. LCC has subsequently created a bin for 'Technology'.
If the Universe is small, like subjects available for the Leaving Certificate, it's easy enough to have a fudge and file Home Economics under both Applied Science and Social Studies. Although, of course the physical book has to go on one shelf or the other: no library is going to buy two copies of All In The Cooking The HomeEc textbook because of a classification problem. And the librarian has to point the punter at the right shelf; at least until everything is transferred to Kindle. But with larger datasets you have to make a choice and that is one of the fundamental skill-sets of librarians.
Philosophically there is a difference. Dewey imagined that he was setting up a system to classify all knowledge (the little we knew then) whereas the LCC is just a catalog of what's in the LoC. That is pretty stonking big - nearly 25 million books and 15 million serials, newspapers, monographs and sheet music. With Google's plan to scan all the books in all the libraries of the world, this heap of writing is now searchable on-line. What do we know that isn't written down? Is it possible to classify [make sense of; put into context; relate to other data] it? Not so much? So the universe of knowledge to a degree is the contents of the LoC - with a little help from other very large libraries.
To a certain extent the war between DDC and LCC is artificial and superficial but so much is invested in both systems now that it would be hard to merge them or ditch one in favour of the other. A researcher at LoC in the 60s developed MARC a digital electronic cataloging system which made the life of regional librarians much easier - they could use MARC wholesale. That could have given LCC a VHS vs Betamax or EBCDIC vs ASCII advantage to sweep Dewey into oblivion but it ain't happened yet. So we're stuck with both and you'll need a conversion chart:
|000||QA, A||Computers, gen. knowledge|
|100||B||Psychology & Philosophy|
|300||H, J, K, L||Social sciences|
|500||Q||Science and Maths|
|600||T, R, S||Technology, Appl Science|
|900||G, D, F||Geography & History|