What is the difference between a subnet mask and a netmask?

Is there any logic in using two different names for determining host ID and net ID?

For example, if you type print route in a Command Prompt you get things with netmask, but the IPv4 settings seems to use subnet mask.

Is there any significant difference between the two terms?

The difference is very, very slight. 9 times out of 10, they will mean the exact same thing.

However, the terms can have a contextual meaning in cases where we're discussing the subnetting of a given network. In those cases, the two terms "network mask" and "sub-network mask" can have distinct meanings. That is, if we make a distinction between a "network" and a "sub-network" then "the mask of a network" and "the mask of a sub-network" mean different things because of the context. This distinction is a relative distinction.

For example, let's say you've been issued the 10.10.0.0/16 network (using CIDR notation). Here, your "network mask" is 255.255.0.0. Let's say you need to separate this network into 4 smaller networks, each as large as they could possibly be. In order to get 4 networks out of 10.10.0.0/16, you need to borrow two bits (00, 01, 10, 11) from the host address and use them for the subnet addresses. This will give you the following sub-networks:

10.10.0.0/18
10.10.64.0/18
10.10.128.0/18
10.10.192.0/18

Here, your "network mask" is still 255.255.0.0, but each "subnet mask" is 255.255.192.0.

But, as I said, it's completely a relative term based on context. One could also talk about 255.255.192.0 being a "network mask" and then 255.255.0.0 being a "supernet mask" if in the same context we're talking about 10.10.0.0/16 being a supernet of, say, 10.10.64.0/18. It's all based on the context of what is being discussed.

The "Netmask", "subnet mask", or simply "mask" are all the same thing: A mask that tells software which IPs belong to that network an which don't.