The "small player" will probably be version 6.4 and the one you've downloaded version 7 or above. (to get which version it is, when you're in media player (both types) click help, about = gives you version number)
As you've got version 6.4 with the os I assume that you will have windows 98.
The later version of media player does indeed use more resources, which may be the skipping you hear. There are things you can do to try and lessen what's going on in the background while the computer is on to free up some resources, like reducing the start up list for eg. If you want to try this you'll have to post you specs/os etc.
Have you also selected the older version to be the player that plays your type of saved music file?
It sounds like the newer version is the player that plays these by default. You need to go into the options of both media players. In the options you can select which type of files each media player plays by default. So if you files are mp3 for example, you need to unselect mp3 from the new player, and select it in the older one (these are in the formats tab in media player options).
Then your older player will be the one that plays that type of file.
You may have to do this a couple of times as the new player likes to keep hold of it's file types sometimes.
Even then I am not sure that a list you created in the new player then export, will play in the old player, my guess is no.
Your other option is to download another media player. There are lots of free, low resource hogging players out there. Winamp is one but if you do a search in this site for free media players, you'll get lots of discussions on which is best.
With this option you will be able to get all the functions of the newer player you have, but shouldn't affect your system or surfing, because they take up far less resources.
You can then create your playlists in that and play them in that without exporting/saving etc.
Shout if you want more.