The 'seed words' do always work.
Are you sure you have the correct backup ? It is weird that you can't recover your wallet on another computer. This should never happen.
Make sure to make a backup (or multiple) of your wallet file and install the latest version of electrum (only from https://electrum.org/).
Afterwards make sure to verify the signature before installing!
Then you should already be able to access your wallet, if not, move the backed up wallet file into the wallet directory.
Yes, because you are using a 2FA wallet (which basically is a 2-of-3 multisignature wallet).
Your wallet does hold 1 key, TrustedCoin (a company which provides this service via 2FA) holds another key.
And your backup seed holds the 3rd key + the key your wallet already holds.
You need 2 out of 3 keys to create a transaction, this means, one of these:
- Your wallet file + 2FA
- Your backup seed