Assigned gender is the gender one is assigned at birth, typically based off of their birth genitalia. Typically one is assigned one of the two primary binary genders most commonly used in western society: male or female. This often leads to the common misassociation of penis with male and vaginas with female. This also perpetuates the false understanding that there are only two genders, when there are in fact, much much more (for more information about the hundreds of genders that exist feel free to explore the rest of this wiki!). The difference between this (assigned gender) and assigned sex is sex is more about genitalia and gender is more of a social construct, and a feeling, who you feel you are deep down in your heart and how you identify yourself to the world.
The fact that we as a society (at least western society) commonly associate gender as a two option male or female thing were you are your gender and you are assigned it strictly based on genitalia alone creates an enroment which leads to gender dysphoria. Also it makes people think that if they do things differently than what people born with that genitalia typically do or think or feel then that makes them a freak so they start to repress their true feelings and their true selves to fit in to our society's cookie cutter definitions of the genders male and female. It's sad to think that many have lived their whole lives without exploring their identity or if they did found their gender was different than the one they were assigned because of their genitalia they couldn't express it because society wouldn't accept they way they truly are. If we all feel free and comfortable to explore ourselves than that fear of being judged for being different will start to dissipate. Also the fact that typically in western society one is usually assigned either male or female makes all other genders, binary and non-binary and agender, seem like they don't exist, wich, for people who don't identity as the typical male-female genders or they are outside the gender spectrum, can lead them to thinking that they don't belong in this world or that they must be crazy. It can also lead to a lot of bullying among their peers, of one is so bold as to express their true gender anyways, and people tease them, misgender them, ignore them, tell them that they are crazy and they need to get help because there are only two genders get over it, you're a girl/boy. Stop being delusional" and even sometimes physical bullying like pushing them or discrimination ( for example "I don't want to be in a group with [them] they are delusional". This discrimination only grows when they get older, as many businesses still do not have an "other" option on a lot of applications and if you do get hired your coworkers and customers my constantly use the wrong pronouns, even if it's as simple as they/them pronouns.
Now you are probably wondering what a good solution for this would be. The solution is assume babies are gender neutral at birth and let them figure out their gender as they explore themselves and grow. Instead of having a gender part on the birth certificate, you can just put "birth genitalia" (I'd say leave genitalia off entirely, but I realize that it might be necessary to know for medical reasons). If we do this, kids won't feel pressured to fall into stereotypical cookie cutter genders and the bullying and discrimination will dissipate.We will live in a world that's less judgmental and more open to individually. where everyone can be who they are deep down and learn this though a prosses of self-exploration. That is why gender neutral pronouns for babies is the best opinion.