Almost impossible to answer, unfortunately. The SATA standards for "normal" SATA do not include a few features that are required in a true eSATA port. Among these are support for longer data cables (using stronger signals) and support for Hot Swapping. Some mobo makers (that is, the chips they use to provide the SATA mobo "Standard" ports) actually have those eSATA extra features included, and some have a few of them but not all, and some have no extra features. When you use the adapter cable alone, you rely entirely on the mobo chips to provide some features that eSATA MAY need to work, but you can't tell whether they are there or not. IF you want to be absolutely sure, you need a true eSATA controller, which that PCI-E board will provide.
You should know, however, that it is common for some eSATA features to be in "Standard" SATA mobo ports. Even if they are missing, sometimes it does not matter. For example, if your cable to the external drive is not too long, the signals may be quite good enough to work reliably even if they are not extra-boosted to eSATA standards. If you do not have Hot Swap support it does not matter if you never try to use that capability, anyway. So many people will try the free (or at least, cheap) option of the adapter cable. If it fails, you wasted very little money finding out you need something better. Be aware, however, that you should NOT try "Hot Swapping" in this case. That is, do not simply disconnect or connect the external drive while the computer is in use. You may corrupt data on the external drive doing that, and that could ruin ALL the data on the drive.