People incorrectly believe that the men's rights movement is a misogynistic hate movement that opposes women's rights and gender equality, when it isn't.

People also lump all of the "Manosphere" together, when there are distinct and often incompatible/opposing currents within it.

The "Manosphere" includes Men's Rights Activists (MRAs), Men Going Their Own Way (MGTOW), Pick-Up Artists (PUAs), involuntary celibates (incels), and the Red Pill Movement (TRP).

MRAs want to change the system, MGTOW wants to leave the system, PUAs and the Red Pill Movement want to take advantage of the system, and incels believe nothing can be done about the system.

MRAs fight for men's rights, and want to change laws, policies, culture, language, and people's beliefs, attitudes, and behavior. 

MGTOW says that men need to leave the system by not having romantic (and sometimes also sexual) relationships, and not having children.

TRP evolved from the PUA community and uses its tactics as a foundation for a broader, more ideological framework. PUAs' scope is more limited to dating and sexual strategy, often using specific psychological and social tactics to attract and coerce women into sexual encounters. On the other hand, TRP extends beyond dating advice to broader social and political commentary. 

Incels believe that they are trapped in celibacy due to society and dating being stacked against them, and typically blame women for this. Incels believe they are entitled to sex.

Most incels are fatalistic, and believe there is no way out of their "involuntary celibacy", and often believe society cannot be changed, and that incels just have to accept their "involuntary celibacy."

These parts of the "Manosphere" differ in their goals, methods, and what they prioritize. In their "pure" forms, they are fundamentally incompatible with each other (except PUAs and the Red Pill Movement), though they can borrow ideas from each other, and ideas can be combined. For example, someone could be an MRA and also subscribe to MGTOW, or to the Red Pill Movement, or be a PUA. Also, many men in the Red Pill Movement are PUAs as well. Also, someone could subscribe to MGTOW and be a PUA, if they only wanted casual sexual encounters but rejected romantic relationships. However, someone couldn't be both part of the MGTOW movement and the Red Pill Movement. Also, incel ideology is fundamentally incompatible with all of the other "Manosphere" currents.

In terms of extremism, from least to most, the order would be:

MRAs

MGTOW

PUAs / the Red Pill Movement

Incels

MRAs have the least problems with extremists (though there are some), MGTOW in some forms can be extremist, PUAs and the Red Pill Movement have a pervasive extremism problem, and Incels are full-blown extremist.

Likewise, I would say the order is the same in terms of misogyny.

Most MRAs aren't misogynists, MGTOW can be misogynistic, PUAs / the Red Pill Movement tend to be very misogynistic, and Incels are extremely misogynistic.

I think part of the reason for the MRM's (undeserved) bad reputation is because people lump it in with the more problematic parts of the "Manosphere".