Do you still have the box that the new switch came in? I wonder if you can post a picture or just post the product name. It sounds as if the old switch fixture had two completely separate switches in it whereas the new has the input side of both switches shorted. Well, come to think of it: for it to work as you describe, you'd also need to install the switch wrong side up - i.e. the inputs would become outputs and vise versa. The switch itself would not care unless of course there's an internal bridge on the input side but its operation would change much as you describe.
Anyhow, see if you can post more information and carefully check for any difference between the old switch and the new - I have a feeling they are not exact same fixtures and have a small but significant internal connection difference (see above)