Time Machine Back Up

Will a Time Machine Backup from one computer installed onto another overwrite all the information on the new computer?

I got a new computer and in order for me to install a Time Machine Backup onto it I have to format the external harddrive I was using. The External HD already had 150/500 GB used on it so I transfered that 150GB of data onto the new computer - a fresh SSD with 750GB of room. The old computer I'm using only has 250GB of storage and over 150GB of that is already in use, so it doesn't have enough room on it to take the data from the External Drive. So, will doing a Time Machine Backup of the old computer over write the 150gb worth of data on the new one?

I hope that made sense...