Yes, grandtanteJE65, the two expressions do mean the same thing, but using the ordinal number (such as thirty-fifth anniversary) rather than the cardinal (thirty-five year anniversary) sounds a lot quicker and less unwieldy to me. Also, if it's not broke...
There seems to be a tendency to put extra words into all sorts of expressions. My bugbears (which I have mentioned before) are horse riding, ice skating and parking up, which used to be known as riding, skating and parking. Royal Mail has now given us another one with its stamp swap -or swap out as it prefers to call it!!