I find that all too often these kinds of questions stem from people trying to translate a word into English rather than investigating the nuances in a J-J dictionary. Borrowing from snailplane's advice, we will avoid translation into English and look at the nuances as explained directly in Japanese.
For まとも（に）, we see the following definition:
(2)道理にかなっていて、他人から非難される点のないこと。きちんとしていて、いかがわしい点のないこと。また、そのさま。
「―な商売」「挨拶すら―にできない」
See an alternate definition here.
This is a word that deals with correctness or fairness. It refers to something that is logical, something that cannot be criticized, something that is done carefully and with nothing fishy about it. It's proper in the sense that it is. If you go from the second link, it basically describes a process that is done straight and with no monkey business. No bullshit, just get it done. This isn't the kind of meaning that you normally associate with something as simple as inspecting a car, and especially in your example sentence there's no real complicated meaning going on; it just refers to checking your car before you drive. If there were some well-established social procedure involved in checking your car, and it was imperative to some sort of business venture or something, or otherwise connected to something for which other people are depending on you, then maybe you could say まともに. But you just want to say to check your car carefully, not with this sense of cosmic correctness. If you look at the examples in the definition, it includes business and greetings. If business is done まともに, you can expect that it's by the books and a smooth process. If someone can't even give a proper aisatsu, though, you know they're probably messing a lot of stuff up beyond just saying hello. Also of note is that there is another definition for this word that involves facing someone or something squarely, so you might be able to think of the more metaphorical meaning of this as an extension of the act of looking someone in the eye, or something along those lines.
Now, if we look up 念入り on the other hand:
細かい点までよく注意すること。念を入れること。また、そのさま。入念。
「―な仕事ぶり」「―に点検する」
It's exactly what you're looking for, and even the example sentence includes 念入りに点検する. The meaning isn't nearly as ambitious: it's just paying attention to details. You could probably translate your example to "careful inspection" in English.
The moral of this story is that if you are trying to translate two very nuanced words into English and compare them based on their English translations then you are easily setting yourself up for confusion. Use Japanese dictionaries as much as you can.
