As long as your 24-pin connector has plugged in fine, then we know it is ATX standard. Many OEM machines have proprietary connectors, though usually they stick to ATX standard for the CPU power.
The 4-pin motherboard is compatible with an 8-pin EPS connector if there is enough room for it, and it looks like there is in either direction. The four pins that match will slide in with the extra 4 just sitting there connected to nothing. Do not force it, it will only line up with the correct 4-pins. Many 8-pin connectors are also 4+4 designed to be split apart, usually they slide together on small rails, or just have a little lip to help alignment.