There isn't much you can do if his mother is sitting on his shoulder and whispering that you are not his "real" family, only she is into his ear all the time.

My ex-SIL managed to alienate my brothers kids from him by telling them that their divorce was between him and them and that all of us on their father's side were no longer part of their family.

As much as it hurts you have to just keep doing what you are doing and try to not let it affect your plans or lives. 

He is a teenager- and they are moody and often anti-social with family by nature. So toss some of it up to that. The rest is BM whispering in his ear that he isn't real family to you and that he shouldn't feel any obligations towards your family. This is why blending families can suck the big one most of the time! 

You can't make him change his mind; he has to do that on his own. But, here are a few things to consider that might help you:

1.) He may not enjoy doing things with his mom, but she doesn't give him a choice. My SSs were in this boat. They didn't want to do anything on weekends with us, partly because they're homebodies and partly because BM dragged them around everywhere during her weeks and weekends. This occurred even when they were teens. Not wanting to do things had nothing to do with us and everything to do with them being peopled-out. Doing things with BM wasn't always a "want to" but a "forced to". 

2.) He is a teenager, and teenagers are fickle creaturea even at the best of times. He may give BM similar problems that you just don't hear about. 

3.) Not all older kids are interested in being role models for younger kids, even siblings/stepsiblings. It's not fair to put SS in that role when he himself is still a child. This is an excellent opportunity to teach your son about boundaries, and about how we can't force ourselves onto other people, even if we care about them.

4.) His BM could be trying to alienate him from his dad and family. Or, she may be playing the "woe is me" card if she's single. Or, before your DH and BM split, she could have been the one facilitating family relationships so your SS is used to her and her family being more prominent (this would be especially true if your DH isn't very close to his family). Any of these dynamics will shape how SS views family, and at 14, he's only starting to hit an age where he might see the problems with this (or see how these dynamics impact him).

5.) SS just may not like you guys. I don't mean that offensively. Every person has a different personality, even in the same family. Having kids is an incredibly selfish act, which means we have to be okay if our kids eventually say "I wish you hadn't been the ones that had me." I'm not saying it isn't hurtful and a gut-punch, but as adults, we can't make one of the most selfish decisions a human can make and then only accept obedience and unconditional love from the person we create. We got what we wanted - a person. That person, though, gets to decide at what level they are okay with you making that decision for them. 

My suggestion? Stop reminding him that you love him. He knows. Invite him to places but don't make a big deal about him not going (provided it's something that it's okay to skip out on, like a movie night). Try to incorporate things he likes into stuff you all decide to do as a family. Remember, it's not just about him assimilating into the family you've created. You all have to grow and adapt to him, too. That may mean spending a weekend here and there just staying home. That may mean finding a babysitter for the younger one and doing something more "adult" with the older one. You also just have to accept that he may not be that into family right now (or ever).

If your DH thinks BM is alienating his son or trying to turn SS against him, he should do research on alienation and find tips/tricks on handling that with his son. It's a very different approach that will likely feel cold and unloving (but actually relieves pressure on kids who feel stuck between parents).

Serious kudos to you for being a loving supportive step mom.  This is definitely hardest job I've ever encountered.  
 

May I ask how long you've been in SS life ? Teenagers suck period.  Even if they were once easy going and into you they can change into people we don't even recognize (happend to me).    
 

I would stop pushing the we love you pretty please come with us stuff.  I'd throw out a casual invite each time and let it go. It's hard but you have to respect his stance with you and your son unless he's being disrespectful.  
 

I'm assuming these two boys are unrelated.   Even with half siblings there's no real way to magically make the bond happen (especially with BMs who don't help the situation). My SS is not into his half bro at all.  It hurts when I think about the bond I have with my siblings but I get passed it.  
 

I encourage lots of friend times and closeness with my friends ( My best friends I refer to them as aunt so and so ) with my young child.  I feel we get that closeness from other family and friends.  I'm not sure how old your son is but keeping him busy with other close people is helpful.  It's also a good lesson (that you can make age appropriate) regarding different people and different personalities  . How some of us prefer to be alone and are introverted etc.  

sucks about the bday but hey if his dad gives him the choice to go or not then he has it. In my family too bad SS goes to the party but Ive been a stepmom for a long time and dynamics differ.  
I wouldn't make your child see you putting in all this effort with no return.  Keep it casual and it will be less hurtful for all parties.  GL