Hellsing Ultimate is not the same. Same voice actors, same characters (at first). OVA 1 is very similar to how the anime started, yes. However, anyone who watched the original series will remember how the Hellsing Organization discovered that a group called Millennium was manufacturing the Freak chips. Then, suddenly, the focus of the last 3 or 4 episodes is on some weird vampire that fights Alucard.

The Ultimate series follows the manga much more closely. This shows up more in OVA 2 and will be obviously different by OVA 3.

[SPOILER="Hellsing manga"]Millennium is a Neo-Nazi group of vampires based in Brazil that is trying to start WWIII with a shitload of undead Freaks and ghouls as their army. The production stayed away from the Nazi issue and introduced a poorly written end to the series.[/SPOILER]