The same Minecraft version alone is not enough to log into the modded server. The mod installer, mod versions and dependencies must also be compatible.
Step by step
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Find out the server's Minecraft version, the installer it uses, and the version of its modpack. Create a separate profile in Launcher with this information.
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Use the client package provided by the server owner. However, before moving visual mods running on the client to the server directory, read the mod's requirements.
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If entry is denied, find the first missing mod, bad version, or registry mismatch in the log. Correct one single difference and try again.
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If a clean and compatible profile connects, compare the mod list of the old profile. There is no need to delete the server world or files.
Keep in mind
The Paper plugin does not load in the same way, even if it appears to be the same file type as the Forge or Fabric mod.