If you are verifying passwords, your first choice is Argon2id in all cases.
In the incredibly unlikely event that you can't use Argon2id, then the tradeoffs between Argon2i and Argon2d matter.
Argon2i is intended to prevent side-channel attacks. That is, if you have an unprivileged malicious process on your server, it could use CPU vulnerabilities like Specter to tease out enough details to reverse engineer the password. By randomizing its memory layout a bit, the malicious processes are far less likely to be able to listen in.
This is very important on servers; especially virtual servers running in cloud environments.
Argon2d, however, is intended to prevent optimizing parallel processing in memory constrained environments like on GPUs, FPGAs, and ASICs. The tradeoff here is that general purpose CPU cores tend to have access to a lot of memory, and multiple banks of memory can be mapped at the same time. GPUs have several programmable lanes (not exactly a CPU core; there's less branching allowed) that have access to far less memory at any given time. Switching the mapping of memory banks takes time and makes using a GPU less desirable. A password hashed by Argon2d is far more resistant to offline brute force attacks, because it is designed so that there should be no shortcuts that an attacker can take to run faster than a general purpose CPU.
If you know that you have physical security over your server (only you can touch it), and you have audited every piece of software on that server (not possible with a modern OS due to the vast amount of software on even the most barebones systems), then you can ignore Argon2i.
If you know that your user database will never be leaked, or you are working with cryptocurrencies where nobody knows which inputs will generate the desired output, you can ignore Argon2d. If you're a cryptocurrency designer who still thinks that proof-of-stake means proof-of-work, then you want the memory hardness of Argon2i, to disincentivize the GPU/FPGA/ASIC arms races that only serve to put money into the pockets of those who are already rich. (And the pockets of your electric company...)
If you are like 99.999% of site owners, use Argon2id.